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Ciência e Agrotecnologia

Print version ISSN 1413-7054On-line version ISSN 1981-1829

Ciênc. agrotec. vol.41 no.4 Lavras July/Aug. 2017

http://dx.doi.org/10.1590/1413-70542017414047716 

Agricultural Sciences

Externally studentized normal midrange distribution

Distribuição da midrange estudentizada externamente

Ben Dêivide de Oliveira Batista1  * 

Daniel Furtado Ferreira2 

Lucas Monteiro Chaves2 

1Universidade Federal de São João del-Rei/UFSJ, Departamento de Matemática e Estatística/DEMAT, São João del-Rei, MG, Brasil

2Universidade Federal de Lavras/UFLA, Departamento de Estatística/DES, Lavras, MG, Brasil

ABSTRACT

The distribution of externally studentized midrange was created based on the original studentization procedures of Student and was inspired in the distribution of the externally studentized range. The large use of the externally studentized range in multiple comparisons was also a motivation for developing this new distribution. This work aimed to derive analytic equations to distribution of the externally studentized midrange, obtaining the cumulative distribution, probability density and quantile functions and generating random values. This is a new distribution that the authors could not find any report in the literature. A second objective was to build an R package for obtaining numerically the probability density, cumulative distribution and quantile functions and make it available to the scientific community. The algorithms were proposed and implemented using Gauss-Legendre quadrature and the Newton-Raphson method in R software, resulting in the SMR package, available for download in the CRAN site. The implemented routines showed high accuracy proved by using Monte Carlo simulations and by comparing results with different number of quadrature points. Regarding to the precision to obtain the quantiles for cases where the degrees of freedom are close to 1 and the percentiles are close to 100%, it is recommended to use more than 64 quadrature points.

Index terms: Distribution function; density function; Gauss-Legendre quadrature; Newton-Raphson method; R.

RESUMO

A distribuição da midrange estudentizada externamente foi criada com base nos procedimentos de estudentização de Student e foi inspirada na distribuição da amplitude estudentizada externamente. O amplo uso da amplitude estudentizada externamente em comparações múltiplas também foi uma das motivações para desenvolver esta nova distribuição. Neste trabalho objetivou-se derivar expressões analíticas da distribuição da midrange estudentizada externamente, obtendo a função de distribuição, função densidade de probabilidade, função quantil e geradores de números aleatórios. Essa é uma nova distribuição que os não há relatos na literatura especializada. Um segundo objetivo foi construir um pacote R para obter numericamente as funções mencionadas e torná-las disponíveis para a comunidade científica. Os algoritmos foram propostos e implementados usando os métodos de quadratura Gauss-Legendre e Newton-Raphson no software R, resultando no pacote SMR, disponível para baixar na página do CRAN. As rotinas implementadas apresentaram alta acurácia, sendo verificadas usando simulação Monte Carlo e pela comparação com diferentes pontos de quadratura. Quanto a precisão para se obter os quantis da distribuição da midrange estudentizada externamente para 1 grau de liberdade ou percentis próximo de 100%, é sugerido utilizar mais do que 64 pontos de quadratura.

Termos para indexação: Função de distribuição; função densidade; quadratura Gauss-Legendre; método Newton-Raphson; algoritmo; R

INTRODUCTION

Many problems on statistical investigations are based on studies of sample order statistics. Important cases of the order statistics are the minimum and maximum value of a sample. Among other functions of order statistics, the range and midrange are of special interest here, which correspond to the difference between the maximum and the minimum and to the average from these two extreme values, respectively. Several authors studied and applied this subject (Tippet, 1925; Pearson and Hartley, 1942; Gumbel, 1944, 1946; Wilks, 1948; Yalcin et. al., 2014; Wan et. al., 2014; Barakat et. al., 2015; Bland, 2015; Mansouri, 2015; Li and Mansouri, 2016) and some of these studies are discussed in the sequence.

Tippet (1925) studied the first four moments of the range. Pearson and Hartley (1942) obtained tabulated values of the cumulative probabilities for several range values in small samples (n = 2 to 20) drawn from normal populations. Gumbel (1944, 1946) established the independence of extreme values for large samples from several continuous distribution, as well as the distribution of the range and midrange. Wilks (1948) reviewed several articles relating to the order statistics and suggested several examples of their applications to statistical inference.

Studies related with studentization were initially proposed by Student (1927) and the particular case of the studentized range distribution have been largely used in multiple comparison procedures in different areas of scientific research since the pioneering works in this area (Duncan, 1952, 1955; Tukey, 1949, 1953). The studentized range refers to the random variable defined simply as the range divided by the sample standard deviation, considering that both terms of this ratio are random variables independently distributed and computed in samples drawn from the normal distribution.

Let Y 1, Y 2, …, Y n be the order statistics in a sample of size , that are defined by sorting the original sample variables X 1, X 2, …, X n in increasing order. The sample X 1, X 2, …, X n are drawn from a population with distribution function F(x). The range is defined by W = Y n - Y 1. The cumulative distribution and the probability density functions (cdf and pdf) of the range are the Equations 1 and 2,

FW(w)=nf(z)[F(w+z)F(z)]n1dz,MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaeeaaaaaa aaa8qacaWGgbGcpaWaaSbaaKqaafaapeGaam4vaaWdaeqaaKaaa9qa caGGOaGaam4DaiaacMcacqGH9aqpcaWGUbGcdaWdXbqcaa0daeaape GaamOzaaqcba0daeaapeGaeyOeI0IaeyOhIukapaqaa8qacqGHEisP aKWaajabgUIiYdqcaaKaaiikaiaadQhacaGGPaGaai4waiaadAeaca GGOaGaam4DaiabgUcaRiaadQhacaGGPaGaeyOeI0IaamOraiaacIca caWG6bGaaiykaiaac2fak8aadaahaaqcbauabeaapeGaamOBaiabgk HiTiaaigdaaaqcaaKaamizaiaadQhacaGGSaaaaa@5924@ (1)

fW(w)=n(n1)f(z)f(w+z)× ×[F(w+z)F(z)]n2dz,MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajaaqqa aaaaaaaaWdbiaadAgak8aadaWgaaqcbauaa8qacaWGxbaapaqabaqc aa0dbiaacIcacaWG3bGaaiykaiabg2da9OWaa8qCaKaaa9aabaWdbi aad6gaaKqaa9aabaWdbiabgkHiTiabg6HiLcWdaeaapeGaeyOhIuka jmaqcqGHRiI8aKaaajaacIcacaWGUbGaeyOeI0IaaGymaiaacMcaca WGMbGaaiikaiaadQhacaGGPaGaamOzaiaacIcacaWG3bGaey4kaSIa amOEaiaacMcacqGHxdaTaOqaaKaaajaaywW7caaMf8UaaGzbVlaayw W7cqGHxdaTkmaadmaajaaqpaqaa8qacaWGgbGaaiikaiaadEhacqGH RaWkcaWG6bGaaiykaiabgkHiTiaadAeacaGGOaGaamOEaiaacMcaai aawUfacaGLDbaak8aadaahaaqcbauabeaapeGaamOBaiabgkHiTiaa ikdaaaqcaaKaamizaiaadQhacaGGSaaaaaa@6DBE@ (2)

respectively, as showed in David and Nagaraja (2003) and in Gumbel (1947).

Considering now samples of size n from the normal distribution with standard deviation σ and mean μMathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0Maey icI4SaeSyhHekaaa@3AA0@ , the externally studentized range is defined by the ratio Q=W ′ X=WSMathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGrbGaeyypa0ZaaSaaa8aabaWdbiqabEfapaGbauaaaeaapeGa aeiwaaaacqGH9aqpdaWcaaWdaeaapeGaae4vaaWdaeaapeGaae4uaa aaaaa@3D04@ where W’ = W/σ is the sample standard range and S 2 is an independent and unbiased estimator of σ2, associated with ν degrees of freedom. The cumulative distribution and the probability density functions, according to David and Nagaraja (2003), are given by Equations 3 and 4,

F(q;n,ν)=0nϕ(y)[Φ(xq+y)Φ(y)]n1××f(x;ν)dydx, (3)

f(q;n,ν)=0n(n1)xϕ(y)ϕ(xq+y)××[Φ(xq+y)Φ(y)]n2f(x;ν)dydx, (4)

where ϕ(y) and Ф(y) are the probability density and cumulative distribution functions from a standard normal population evaluated at y, with y ∈]−∞,∞[ and f(x;ν) is the probability density function of X = S/σ. Considering that X is obtained in a sample of size ν+1 from the normal distribution, then it is well known the fact that νS2σ2χν2MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWcaaWdaeaapeGaeqyVd4Maam4ua8aadaahaaWcbeqaa8qacaaI YaaaaaGcpaqaa8qacqaHdpWCpaWaaWbaaSqabeaapeGaaGOmaaaaaa GccqGH8iIFcqaHhpWypaWaa0baaSqaa8qacqaH9oGBa8aabaWdbiaa ikdaaaaaaa@42DA@ , i.e., it has a chi-square distribution with ν degrees of freedom (Mood et al., 1974). Hence, νS2νσ2= S2σ2χν2/νMathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWcaaWdaeaapeGaeqyVd4Maam4ua8aadaahaaWcbeqaa8qacaaI YaaaaaGcpaqaa8qacqaH9oGBcqaHdpWCpaWaaWbaaSqabeaapeGaaG OmaaaaaaGccqGH9aqpcaGGGcWaaSaaa8aabaWdbiaadofapaWaaWba aSqabeaapeGaaGOmaaaaaOWdaeaapeGaeq4Wdm3damaaCaaaleqaba WdbiaaikdaaaaaaOGaeyipI4Naeq4Xdm2damaaDaaaleaapeGaeqyV d4gapaqaa8qacaaIYaaaaOGaai4laiabe27aUbaa@4E3E@ . Therefore, it can be concluded that X=Sσχν2/νMathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGybGaeyypa0ZaaSaaa8aabaWdbiaadofaa8aabaWdbiabeo8a ZbaacqGH8iIFdaGcaaWdaeaapeGaeq4Xdm2damaaDaaaleaapeGaeq yVd4gapaqaa8qacaaIYaaaaOGaai4laiabe27aUbWcbeaaaaa@4390@ .

Theorem 1. If X=S/σ is computed in a sample of size ν+1 from a normal distribution with mean μ and variance σ², then its probability density function is given by Equation 5,

f(x;ν)=νν/2Γ(ν/2)2ν/21xν1eνx2/2,x0.MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGMbGaaiikaiaadIhacaGG7aGaeqyVd4Maaiykaiabg2da9maa laaapaqaa8qacqaH9oGBpaWaaWbaaSqabeaapeGaeqyVd4Maai4lai aaikdaaaaak8aabaWdbiabfo5ahjaacIcacqaH9oGBcaGGVaGaaGOm aiaacMcacaaIYaWdamaaCaaaleqabaWdbiabe27aUjaac+cacaaIYa GaeyOeI0IaaGymaaaaaaGccaWG4bWdamaaCaaaleqabaWdbiabe27a UjabgkHiTiaaigdaaaGccaWGLbWdamaaCaaaleqabaWdbiabgkHiTi abe27aUjaadIhapaWaaWbaaWqabeaapeGaaGOmaaaaliaac+cacaaI YaaaaOGaaiilaiaadIhacqGHLjYScaaIWaGaaiOlaaaa@5F18@ (5)

Proof. If Uχν2MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGvbGaeyipI4Naeq4Xdm2damaaDaaaleaapeGaeqyVd4gapaqa a8qacaaIYaaaaaaa@3CEF@ , as discussed above, then the distribution of X = S/σ can be obtained from the transformation U = v X 2 . The Jacobian of this transformation is given by

J=dudx=2νx,MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGkbGaeyypa0ZaaSaaa8aabaWdbiaadsgacaWG1baapaqaa8qa caWGKbGaamiEaaaacqGH9aqpcaaIYaGaeqyVd4MaamiEaiaacYcaaa a@412A@

for x > 0.

The density function of X, from samples of a normal population, is obtained from the chi-square distribution by

f(x;ν)=fU(u;ν)|J| =12ν/2Γ(ν/2)uν/21eu/2|J|=12ν/2Γ(ν/2)(νx2)ν/21eνx2/22νx=2ννν/212ν/2Γ(ν/2)x(x2)ν/21eνx2/2

resulting in Equation 5.

Another very interesting order statistic is the midrange and introduced, among others, by Gumbel (1947) and Rider (1957).

Definition 1. The midrange is defined as the mean between the minimum and the maximum order statistics by Equation 6,

R¯=(Y1+Yn)/2.MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGsbGbaebacqGH9aqpcaGGOaGaamywa8aadaWgaaWcbaWdbiaa igdaa8aabeaak8qacqGHRaWkcaWGzbWdamaaBaaaleaapeGaamOBaa WdaeqaaOWdbiaacMcacaGGVaGaaGOmaiaac6caaaa@40BA@ (6)

The externally studentized midrange is defined considering the original studentization procedures of Student (1927) and was inspired in the externally studentized range definition.

Definition 2. The externally studentized midrange are defined by

Q¯=R¯/S,MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGrbGbaebacqGH9aqpceWGsbGbaebacaGGVaGaam4uaiaacYca aaa@3B35@

where R¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGsbWdayaaraaaaa@3715@ is expressed in Equation 6 and S is an estimator of the population standard deviation σ with ν degrees of freedom obtained independently from R¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGsbWdayaaraaaaa@3715@ . It should be noticed that Q¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGrbWdayaaraaaaa@3714@ is a random variable.

However, few studies address the midrange distribution and none was found on the externally studentized midrange, considering normal or non-normal populations. The importance of studies about the distribution of the externally studentized midrange could be enormous in the analysis of experiments. Rider (1957), among others, proved that the midrange estimator is more efficient than the sample mean in platykurtic distributions. Another important aspect that could be useful is the proposition of multiple comparison procedures based on externally studentized midrange, that could potentially show better results than the traditional tests based on the studentized range.

This work aimed to obtain the externally studentized midrange distribution. It intended to develop analytic equations to distribution of Q¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGrbWdayaaraaaaa@3714@ , obtaining the cumulative distribution, probability density and quantile functions and generating random values. A second objective was to build an R package (R Development Core Team, 2017) for obtaining numerically the probability density, cumulative distribution and quantile functions using Gaussian quadratures and Newton-Raphson method and make the R package available to the scientific community.

MATERIAL AND METHODS

The externally studentized normal midrange distribution

Let Y 1, Y 2, … , Y n be the order statistics in a sample of size n, that are defined by sorting the original sample variables X 1, X 2, … , X n , in increasing order. The sample X 1, X 2, … , X n are drawn from a population with distribution function F(x). Therefore, the distribution of the midrange is given by the following theorem.

Theorem 2. The probability density function and the cumulative distribution function of R¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGsbWdayaaraaaaa@3715@ , Definition 1, from a random sample X1, X2, … , Xn, of size n, where Xj has distribution function F(x) and probability density function f(x), j = 1, 2, … , n given by Equations 7 and 8,

fR¯(r¯)=r¯2n(n1)f(z)f(2r¯z)××[F(2r¯z)F(z)]n2dz, (7)

and

FR¯(r¯) =nr¯f(z)[F(2r¯z)F(z)]n1dz,MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaeeaaaaaa aaa8qacaWGgbGcpaWaaSbaaKqaafaapeGabmOuayaaraaapaqabaqc aa0dbiaacIcaceWGYbGbaebacaGGPaWdaiaqbccapeGaeyypa0Jaam OBaOWaa8qmaKaaa9aabaWdbiaadAgaaKqaa9aabaWdbiabgkHiTiab g6HiLcWdaeaapeGabmOCayaaraaajmaqcqGHRiI8aKaaajaacIcaca WG6bGaaiykaiaacUfacaWGgbGaaiikaiaaikdaceWGYbGbaebacqGH sislcaWG6bGaaiykaiabgkHiTiaadAeacaGGOaGaamOEaiaacMcaca GGDbGcpaWaaWbaaKqaafqabaWdbiaad6gacqGHsislcaaIXaaaaKaa ajaadsgacaWG6bGaaiilaaaa@5A85@ (8)

respectively

Proof. Let the joint distribution of Y 1 and Y n (David and Nagaraja, 2003) given by Equation 9,

fY1,Yn(u,x)=n(n1)f(u)f(x)××[F(x)F(u)]n2, (9)

for u< x, then to obtain the distribution of R¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGsbWdayaaraaaaa@3715@ , the transformations Y1=g11(R¯ , Z)= ZMathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGzbWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiabg2da9iaa dEgapaWaa0baaSqaa8qacaaIXaaapaqaa8qacqGHsislcaaIXaaaaO WaaeWaa8aabaWdbiqadkfapaGbaebapeGaaiiOaiaacYcacaGGGcGa amOwaaGaayjkaiaawMcaaiabg2da9iaacckacaWGAbaaaa@4684@ and Yn=g21(R¯ , Z)= 2R¯ZMathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGzbWdamaaBaaaleaapeGaamOBaaWdaeqaaOWdbiabg2da9iaa dEgapaWaa0baaSqaa8qacaaIYaaapaqaa8qacqGHsislcaaIXaaaaO WaaeWaa8aabaWdbiqadkfapaGbaebapeGaaiiOaiaacYcacaGGGcGa amOwaaGaayjkaiaawMcaaiabg2da9iaacckacaaIYaGabmOua8aaga qea8qacqGHsislcaWGAbaaaa@4974@ should be considered. The Jacobian of these transformations is

J=|y1zy1r¯ynzynr¯|=|1012|=2.MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaeeaaaaaa aaa8qacaWGkbGaeyypa0Jcdaabdaqcaa0daeaafaqabeGacaaabaGc peWaaSaaaKaaa9aabaWdbiabgkGi2kaadMhak8aadaWgaaqcbauaa8 qacaaIXaaapaqabaaajaaqbaWdbiabgkGi2kaadQhaaaaapaqaaOWd bmaalaaajaaqpaqaa8qacqGHciITcaWG5bGcpaWaaSbaaKqaafaape GaaGymaaWdaeqaaaqcaauaa8qacqGHciITceWGYbGbaebaaaaapaqa aOWdbmaalaaajaaqpaqaa8qacqGHciITcaWG5bGcpaWaaSbaaKqaaf aapeGaamOBaaWdaeqaaaqcaauaa8qacqGHciITcaWG6baaaaWdaeaa k8qadaWcaaqcaa0daeaapeGaeyOaIyRaamyEaOWdamaaBaaajeaqba Wdbiaad6gaa8aabeaaaKaaafaapeGaeyOaIyRabmOCayaaraaaaaaa aiaawEa7caGLiWoacqGH9aqpkmaaemaajaaqpaqaauaabeqaciaaae aapeGaaGymaaWdaeaapeGaaGimaaWdaeaapeGaeyOeI0IaaGymaaWd aeaapeGaaGOmaaaaaiaawEa7caGLiWoacqGH9aqpcaaIYaGaaiOlaa aa@6422@

Therefore, the joint density of R¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGsbWdayaaraaaaa@3715@ and Z, using 9, is given by

fR¯,Z(r¯,z)= |J|fY1,Yn(g11(r¯,z),g21(r¯,z))=2n(n1)f(z)f(2r¯z)××[F(2r¯z)F(z)]n2.

The required density fR¯(r¯)MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGMbWdamaaBaaaleaapeGabmOua8aagaqeaaqabaGcpeWaaeWa a8aabaWdbiqadkhapaGbaebaa8qacaGLOaGaayzkaaaaaa@3B3B@ can be obtained by integrating the above joint density in relation to z, resulting in

fR¯(r¯)=r¯2n(n1)f(z)f(2r¯z)××[F(2r¯z)F(z)]n2dz.

The distribution function R¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGsbWdayaaraaaaa@3715@ can be obtained by Equation 10,

FR¯(r¯)=r¯t2n(n1)f(z)f(2tz)××[F(2tz)F(z)]n2dzdt. (10)

The simplification of the equation 10 can be performed by inversion of the order of the integrals. There is a dependency between R¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGsbWdayaaraaaaa@3715@ and Z, then by fixing z, t will vary in the interval [ z,r¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEaiaacY caceWGYbGbaebaaaa@38B5@ ]. Note that the upper limit of the interval, r¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGYbWdayaaraaaaa@3735@ , refers to the definition of the cumulative distribution function. Therefore, the results of the change of integration order is

FR¯(r¯)=r¯[zr¯2n(n1)f(z)f(2tz)××[F(2tz)F(z)]n2dt]dz.

Note that

ddt[F(2tz)F(z)]n1=2(n1)f(2tz)××[F(2tz)F(z)]n2.

So,

FR¯(r¯)=r¯nf(z){[F(2tz)F(z)]n1}t=zt=r¯dz.MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaeeaaaaaa aaa8qacaWGgbGcpaWaaSbaaKqaafaapeGabmOuayaaraaapaqabaqc aa0dbiaacIcaceWGYbGbaebacaGGPaGaeyypa0JcdaWdXaqcaa0dae aapeGaamOBaaqcba0daeaapeGaeyOeI0IaeyOhIukapaqaa8qaceWG YbGbaebaaKWaajabgUIiYdqcaaKaamOzaiaacIcacaWG6bGaaiykaO WaaiWaaKaaa9aabaWdbiaacUfacaWGgbGaaiikaiaaikdacaWG0bGa eyOeI0IaamOEaiaacMcacqGHsislcaWGgbGaaiikaiaadQhacaGGPa GaaiyxaOWdamaaCaaajeaqbeqaa8qacaWGUbGaeyOeI0IaaGymaaaa aKaaajaawUhacaGL9baak8aadaqhaaqcbauaa8qacaWG0bGaeyypa0 JaamOEaaWdaeaapeGaamiDaiabg2da9iqadkhagaqeaaaajaaqcaWG KbGaamOEaiaac6caaaa@631A@

As,

{[F(2tz)F(z)]n1}t=zt=r¯=[F(2r¯z)F(z)]n1[F(2zz)F(z)]n1=[F(2r¯z)F(z)]n1,

the result of the cumulative distribution function fR¯(r¯)MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGMbWdamaaBaaaleaapeGabmOua8aagaqeaaqabaGcpeWaaeWa a8aabaWdbiqadkhapaGbaebaa8qacaGLOaGaayzkaaaaaa@3B3B@ is

FR¯(r¯)=nr¯f(z)[F(2r¯z)F(z)]n1dz,MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaeeaaaaaa aaa8qacaWGgbGcpaWaaSbaaKqaafaapeGabmOuayaaraaapaqabaqc aa0dbiaacIcaceWGYbGbaebacaGGPaGaeyypa0JaamOBaOWaa8qmaK aaa9aabaWdbiaadAgaaKqaa9aabaWdbiabgkHiTiabg6HiLcWdaeaa peGabmOCayaaraaajmaqcqGHRiI8aKaaajaacIcacaWG6bGaaiykai aacUfacaWGgbGaaiikaiaaikdaceWGYbGbaebacqGHsislcaWG6bGa aiykaiabgkHiTiaadAeacaGGOaGaamOEaiaacMcacaGGDbGcpaWaaW baaKqaafqabaWdbiaad6gacqGHsislcaaIXaaaaKaaajaadsgacaWG 6bGaaiilaaaa@5982@

as showed in Gumbel (1958).

RESULTS AND DISCUSSION

In the particular case of standard normal population, the Equations 7 and 8 can be rewritten by Equations 11 and 12,

fR¯(r¯)=r¯2n(n1)ϕ(z)ϕ(2r¯z)××[Φ(2r¯z)Φ(z)]n2dz, (11)

and

FR¯(r¯)=nr¯ϕ(z)[Φ(2r¯z)Φ(z)]n1dz,MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaeeaaaaaa aaa8qacaWGgbGcpaWaaSbaaKqaafaapeGabmOuayaaraaapaqabaqc aa0dbiaacIcaceWGYbGbaebacaGGPaGaeyypa0JaamOBaOWaa8qmaK aaa9aabaWdbiabew9aMbqcba0daeaapeGaeyOeI0IaeyOhIukapaqa a8qaceWGYbGbaebaaKWaajabgUIiYdqcaaKaaiikaiaadQhacaGGPa Gaai4waiabfA6agjaacIcacaaIYaGabmOCayaaraGaeyOeI0IaamOE aiaacMcacqGHsislcqqHMoGrcaGGOaGaamOEaiaacMcacaGGDbGcpa WaaWbaaKqaafqabaWdbiaad6gacqGHsislcaaIXaaaaKaaajaadsga caWG6bGaaiilaaaa@5BBD@ (12)

respectively.

If samples from a normal distribution with mean 0 and variance σ2 are considered, then the distribution of R¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGsbWdayaaraaaaa@3715@ will depend on σ. The cumulative distribution function of R¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGsbWdayaaraaaaa@3715@ in this case is obtaining directly from 12 by Equation 13,

FR¯(r¯)=nr¯ϕ0,σ2(z)[Φ0,σ2(2r¯z)Φ0,σ2(z)]n1dz,MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaceaaaaaa aaa8qacaWGgbGcpaWaaSbaaKqaGeaapeGabmOuayaaraaapaqabaqc aaYdbiaacIcaceWGYbGbaebacaGGPaGaeyypa0JaamOBaOWaa8qmaK aaG8aabaWdbiabew9aMPWdamaaBaaajeaibaWdbiaaicdacaGGSaGa eq4Wdm3cpaWaaWbaaKGaGeqabaWdbiaaikdaaaaajeaipaqabaaaba WdbiabgkHiTiabg6HiLcWdaeaapeGabmOCayaaraaajmaicqGHRiI8 aKaaGiaacIcacaWG6bGaaiykaiaacUfacqqHMoGrk8aadaWgaaqcba saa8qacaaIWaGaaiilaiabeo8aZTWdamaaCaaajiaibeqaa8qacaaI YaaaaaqcbaYdaeqaaKaaG8qacaGGOaGaaGOmaiqadkhagaqeaiabgk HiTiaadQhacaGGPaGaeyOeI0IaeuOPdyKcpaWaaSbaaKqaGeaapeGa aGimaiaacYcacqaHdpWCl8aadaahaaqccasabeaapeGaaGOmaaaaaK qaG8aabeaajaaipeGaaiikaiaadQhacaGGPaGaaiyxaOWdamaaCaaa jeaibeqaa8qacaWGUbGaeyOeI0IaaGymaaaajaaicaWGKbGaamOEai aacYcaaaa@69FD@ (13)

where ϕ0,σ 2 (z) and Ф0,σ 2 (z) are the probability density and cumulative distribution functions, respectively, of the normal distribution with mean 0 and variance σ2. The probability density function Ф0,σ 2 (z) is related to the probability density function of the standard normal distribution by ϕ0,σ 2 (z) = ϕ(z/σ)/σ. In the same way, the relationship between the cumulative probability functions is ϕ0,σ 2 (z) = ϕ(z/σ). Hence, if Z/σ is denote by Y, the cumulative distribution function, Equation 13, can be rewritten by Equation 14,

FR¯(r¯/σ)=nr¯/σ1σϕ(y)[Φ(2r¯/σy)Φ(y)]n1dy.MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaceaaaaaa aaa8qacaWGgbGcpaWaaSbaaKqaGeaapeGabmOuayaaraaapaqabaqc aaYdbiaacIcaceWGYbGbaebacaGGVaGaeq4WdmNaaiykaiabg2da9i aad6gakmaapedajaaipaqaaOWdbmaalaaajaaipaqaa8qacaaIXaaa paqaa8qacqaHdpWCaaaajeaipaqaa8qacqGHsislcqGHEisPa8aaba Wdbiqadkhagaqeaiaac+cacqaHdpWCaKWaGiabgUIiYdqcaaIaeqy1 dyMaaiikaiaadMhacaGGPaGaai4waiabfA6agjaacIcacaaIYaGabm OCayaaraGaai4laiabeo8aZjabgkHiTiaadMhacaGGPaGaeyOeI0Ia euOPdyKaaiikaiaadMhacaGGPaGaaiyxaOWdamaaCaaajeaibeqaa8 qacaWGUbGaeyOeI0IaaGymaaaajaaicaWGKbGaamyEaiaac6caaaa@64FC@ (14)

Similarly, the same procedure can be realized for Equation 11.

Definition 3. The standardized midrange is defined by Equation 15,

W¯=R¯σ,MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbGbaebacqGH9aqpdaWcaaWdaeaapeGabmOuayaaraaapaqa a8qacqaHdpWCaaGaaiilaaaa@3BC0@ (15)

where R¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGsbWdayaaraaaaa@3715@ is the midrange from Definition 1 and is the population standard deviation.

Theorem 3. The probability density function and cumulative distribution function of W¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4vayaara aaaa@36EB@ , from Equation 15 are given by Equations 16 and 17,

fW¯(w¯)=w¯2n(n1)ϕ(y)ϕ(2w¯y)××[Φ(2w¯y)Φ(y)]n2dy, (16)

and

FW¯(w¯)=nw¯ϕ(y)[Φ(2w¯y)Φ(y)]n1dy,MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaeeaaaaaa aaa8qacaWGgbGcpaWaaSbaaKqaafaapeGabm4vayaaraaapaqabaqc aa0dbiaacIcaceWG3bGbaebacaGGPaGaeyypa0JaamOBaOWaa8qmaK aaa9aabaWdbiabew9aMbqcba0daeaapeGaeyOeI0IaeyOhIukapaqa a8qaceWG3bGbaebaaKWaajabgUIiYdqcaaKaaiikaiaadMhacaGGPa Gaai4waiabfA6agjaacIcacaaIYaGabm4DayaaraGaeyOeI0IaamyE aiaacMcacqGHsislcqqHMoGrcaGGOaGaamyEaiaacMcacaGGDbGcpa WaaWbaaKqaafqabaWdbiaad6gacqGHsislcaaIXaaaaKaaajaadsga caWG5bGaaiilaaaa@5BCD@ (17)

respectively.

Proof. The cumulative distribution function of W¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaraaaaa@371A@ is obtaining from the cumultive distribution function given in 14. Hence, making the transformations of variables W¯=R¯/σMathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4vayaara Gaeyypa0JabmOuayaaraGaai4laiabeo8aZbaa@3B56@ and Y, with the Jacobian J = σ, the cumulative distribution function of W¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaraaaaa@371A@ is given by

FW¯(w¯)=nw¯ϕ(y)[Φ(2w¯y)Φ(y)]n1dyMathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaeeaaaaaa aaa8qacaWGgbGcpaWaaSbaaKqaafaapeGabm4vayaaraaapaqabaqc aa0dbiaacIcaceWG3bGbaebacaGGPaGaeyypa0JaamOBaOWaa8qmaK aaa9aabaWdbiabew9aMbqcba0daeaapeGaeyOeI0IaeyOhIukapaqa a8qaceWG3bGbaebaaKWaajabgUIiYdqcaaKaaiikaiaadMhacaGGPa Gaai4waiabfA6agjaacIcacaaIYaGabm4DayaaraGaeyOeI0IaamyE aiaacMcacqGHsislcqqHMoGrcaGGOaGaamyEaiaacMcacaGGDbGcpa WaaWbaaKqaafqabaWdbiaad6gacqGHsislcaaIXaaaaKaaajaadsga caWG5baaaa@5B1D@

and the probability density function of W¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaraaaaa@371A@ is obtained by dFW¯(w¯)/dw¯=fW¯(w¯)MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGKbGaamOra8aadaWgaaWcbaWdbiqadEfagaqeaaWdaeqaaOWd biaacIcaceWG3bGbaebacaGGPaGaai4laiaadsgaceWG3bGbaebacq GH9aqpcaWGMbWdamaaBaaaleaapeGabm4vayaaraaapaqabaGcpeGa aiikaiqadEhagaqeaiaacMcaaaa@4415@ , that is,

fW¯(w¯)=nw¯ϕ(y)ddw¯[Φ(2w¯y)Φ(y)]n1dy.MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaeeaaaaaa aaa8qacaWGMbGcpaWaaSbaaKqaafaapeGabm4vayaaraaapaqabaqc aa0dbiaacIcaceWG3bGbaebacaGGPaGaeyypa0JaamOBaOWaa8qmaK aaa9aabaWdbiabew9aMbqcba0daeaapeGaeyOeI0IaeyOhIukapaqa a8qaceWG3bGbaebaaKWaajabgUIiYdqcaaKaaiikaiaadMhacaGGPa GcdaWcaaqcaa0daeaapeGaamizaaWdaeaapeGaamizaiqadEhagaqe aaaacaGGBbGaeuOPdyKaaiikaiaaikdaceWG3bGbaebacqGHsislca WG5bGaaiykaiabgkHiTiabfA6agjaacIcacaWG5bGaaiykaiaac2fa k8aadaahaaqcbauabeaapeGaamOBaiabgkHiTiaaigdaaaqcaaKaam izaiaadMhacaGGUaaaaa@5F76@

Solving

ddw¯[Φ(2w¯y)Φ(y)]n1=2(n1)ϕ(2w¯y)××[Φ(2w¯y)Φ(y)]n2,

the probability density function of W¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaraaaaa@371A@ is

fW¯(w¯)=w¯2n(n1)ϕ(y)ϕ(2w¯y)××[Φ(2w¯y)Φ(y)]n2dy,

as expected.

It is worth noting that the probability density functions 11 and 16 and the cumulative probability functions 12 and 17 are equals. Therefore, the standardized midrange distribution obtained in normal populations with mean zero and variance σ2 is the same of the midrange from standard normal populations.

However, the objective is to find the externally studentized normal midrange ( Q¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGrbWdayaaraaaaa@3714@ , Definition 2) distribution, where Q¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGrbWdayaaraaaaa@3714@ can also be defined by Equation 18,

Q¯=R¯/σS/σ=R¯S=W¯X,MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaageaaaaaa aaa8qaceWGrbGbaebacqGH9aqpkmaalaaajaaypaqaa8qaceWGsbGb aebacaGGVaGaeq4Wdmhapaqaa8qacaWGtbGaai4laiabeo8aZbaacq GH9aqpkmaalaaajaaypaqaa8qaceWGsbGbaebaa8aabaWdbiaadofa aaGaeyypa0JcdaWcaaqcaa2daeaapeGabm4vayaaraaapaqaa8qaca WGybaaaiaacYcaaaa@47BD@ (18)

where W¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaraaaaa@371A@ and S, with v degrees of freedom, are independently distributed. This occurs, for example, when W¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaraaaaa@371A@ is obtained from a random sample of size n from a normal population, and S, the standard deviation, is from another random sample of size v + 1. The independence also occurs when W¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaraaaaa@371A@ is a function of factor means in an experimental design, with n levels and S=MSE/rMathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGtbGaeyypa0ZaaOaaa8aabaWdbiaad2eacaWGtbGaamyraiaa c+cacaWGYbaaleqaaaaa@3C4C@ , where MSE is the mean square error with v degrees of freedom and r is the number of replications associated with each treatment mean, as can be found, e.g., in Searle (1987).

Theorem 4. The probability density function and cumulative distribution function of Q¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGrbWdayaaraaaaa@3714@ from Definition 2 are given by Equations 19 and 20,

f(q¯;n,ν)=0xq¯2n(n1)xϕ(y)ϕ(2xq¯y)××[Φ(2xq¯y)Φ(y)]n2f(x;ν)dydx, (19)

and

F(q¯;n,ν)=0xq¯nϕ(y)[Φ(2xq¯y)Φ(y)]n1××f(x;ν)dydx, (20)

respectively.

Proof. The distribution of Q¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGrbWdayaaraaaaa@3714@ given in (18), is obtained from the joint distribution of W¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaraaaaa@371A@ given in 16 and X = S/σ given in 5. As the variables W¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaraaaaa@371A@ and X are independently distributed, the joint density is the product of their marginal densities, given by Equation 21,

f(w¯,x;n,ν)=w¯2n(n1)ϕ(y)ϕ(2w¯y)××[Φ(2w¯y)Φ(y)]n2f(x;ν)dy, (21)

where f(x;y) is given in 5.

Considering the transformations Q¯=W¯/XMathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyuayaara Gaeyypa0Jabm4vayaaraGaai4laiaadIfaaaa@3A6F@ and X, the Jacobian is

J=|w¯q¯w¯xxq¯xx|=|xq¯01|=x, x>0.MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaeeaaaaaa aaa8qacaWGkbGaeyypa0Jcdaabdaqcaa0daeaafaqabeGacaaabaGc peWaaSaaaKaaa9aabaWdbiabgkGi2kqadEhagaqeaaWdaeaapeGaey OaIyRabmyCayaaraaaaaWdaeaak8qadaWcaaqcaa0daeaapeGaeyOa IyRabm4Dayaaraaapaqaa8qacqGHciITcaWG4baaaaWdaeaak8qada Wcaaqcaa0daeaapeGaeyOaIyRaamiEaaWdaeaapeGaeyOaIyRabmyC ayaaraaaaaWdaeaak8qadaWcaaqcaa0daeaapeGaeyOaIyRaamiEaa WdaeaapeGaeyOaIyRaamiEaaaaaaaacaGLhWUaayjcSdGaeyypa0Jc daabdaqcaa0daeaafaqabeGacaaabaWdbiaadIhaa8aabaWdbiqadg hagaqeaaWdaeaapeGaaGimaaWdaeaapeGaaGymaaaaaiaawEa7caGL iWoacqGH9aqpcaWG4bGaaiilaiaaysW7caWG4bGaeyOpa4JaaGimai aac6caaaa@6259@

Therefore, from the above transformations W¯=XQ¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4vayaara Gaeyypa0Jaamiwaiqadgfagaqeaaaa@39BC@ and using the joint distribution of W¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaraaaaa@371A@ and X, given in 21, the joint distribution of X and Q¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGrbWdayaaraaaaa@3714@ is given by Equation 22,

f(q¯,x;n,v)=f(w¯,x;n,ν)|J|,=xq¯2n(n1)ϕ(y)ϕ(2xq¯y)××[Φ(2xq¯y)Φ(y)]n2f(x;ν)|x|dy. (22)

Integrating the Equation 22 with respect to x, for x[0,]MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG4bGaeyicI4Saai4waiaaicdacaGGSaGaeyOhIuQaaiyxaaaa @3D32@ , the probability density function of Q¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGrbWdayaaraaaaa@3714@ is obtained. Thus,

f(q¯;n,ν)=0xq¯2n(n1)xϕ(y)ϕ(2xq¯y)××[Φ(2xq¯y)Φ(y)]n2f(x;ν)dydx.

The cumulative distribution function Q¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGrbWdayaaraaaaa@3714@ can be obtained by performing the integration of 19 in q¯MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGXbGbaebaaaa@3724@ over [,q¯]MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiabgk HiTiabg6HiLkaacYcaceWGXbGbaebacaGGDbaaaa@3BD3@ . Changing the variable to z, the cumulative distribution function of Q¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGrbWdayaaraaaaa@3714@ is given by

F(q¯;n,ν)=q¯0xz2n(n1)xϕ(y)ϕ(2xzy)××[Φ(2xzy)Φ(y)]n2f(x;ν)dydxdz.

Changing the order of the integrals, and fixing the smallest studentized order statistic, the lower limit with respect to Q¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGrbWdayaaraaaaa@3714@ is now y/x. So,

F(q¯;n,ν)=0xq¯nϕ(y){y/xq¯2x(n1)ϕ(2xzy)××[Φ(2xzy)Φ(y)]n2dz}f(x;ν)dydx.

Since

ddz[Φ(2xzy)Φ(y)]n1=2x(n1)ϕ(2xzy)××[Φ(2xzy)Φ(y)]n2,

then,

F(q¯;n,ν)=0xq¯nϕ(y)[{Φ(2xzy)Φ(y)}n1]z=y/xz=q¯××f(x;ν)dydx.

Solving,

[{Φ(2xzy)Φ(y)}n1]z=y/xz=q¯={Φ(2xq¯y)Φ(y)}n1{Φ(2x.yxy)Φ(y)}n1={Φ(2xq¯y)Φ(y)}n1,

the cumulative distribution function of Q¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyuayaara aaaa@36E5@ is given by

F(q¯;n,ν)=0xq¯nϕ(y)[Φ(2xq¯y)Φ(y)]n1××f(x;ν)dydx,

as expected.

The Equations 19 and 20 are the probability density and cumulative distribution functions, respectively, of Q¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGrbWdayaaraaaaa@3714@ , which is the externally studentized normal midrange. This is a novel distribution that the authors could not find any report in the literature.

In the next section, numerical methods for obtaining the probability density and cumulative distribution functions are described. Also, the quantile functions are also taken into account. The same approaches will be considered for the distribution of W¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaraaaaa@371A@ , Equations 16 and 17, which is the standard normal midrange.

Gauss-Legendre quadrature

The basic idea of Gauss-Legendre quadrature of a function f(x) is to write equal the Equation 23,

11f(x)dx=11w(x)g(x)dxk=1swkg(xk),MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWdXbqcaa2daeaapeGaamOzaaqcba2daeaapeGaeyOeI0IaaGym aaWdaeaapeGaaGymaaqcdaMaey4kIipajaaycaGGOaGaamiEaiaacM cacaWGKbGaamiEaiabg2da9OWaa8qCaKaaG9aabaWdbiaadEhaaKqa G9aabaWdbiabgkHiTiaaigdaa8aabaWdbiaaigdaaKWaGjabgUIiYd qcaaMaaiikaiaadIhacaGGPaGaam4zaiaacIcacaWG4bGaaiykaiaa dsgacaWG4bGaeyisISRcdaaeWbqcaa2daeaapeGaam4DaOWdamaaBa aajeaybaWdbiaadUgaa8aabeaaaeaapeGaam4Aaiabg2da9iaaigda a8aabaWdbiaadohaaKWaGjabggHiLdqcaaMaam4zaiaacIcacaWG4b GcpaWaaSbaaKqaGfaapeGaam4AaaWdaeqaaKaaG9qacaGGPaGaaiil aaaa@6420@ (23)

where f(x) ≡ w(x)g(x) and w(x) is the weight function in the Gaussian quadrature, w k and x k are the nodes and weights, respectively, in an s-point Gaussian quadrature rule, for k = 1, 2, …, s. The weight function is w(x)=1 in the Gauss-Legendre quadrature, thus f(x)=g(x). The set {x k , w k } should be determined such that equation 23 yields an exact result for polynomials of degree 2s-1 or less. For non-polynomial function the Gauss-Legendre quadrature error is defined by Equation 24,

εs|k=1iwkg(xk)k=1swkg(xk)|, i>s.MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaageaaaaaa aaa8qacqaH1oqzk8aadaWgaaqcbawaa8qacaWGZbaapaqabaqcaa2d biabgIKi7QWaaqWaaKaaG9aabaGcpeWaaabCaKaaG9aabaWdbiaadE hak8aadaWgaaqcbawaa8qacaWGRbaapaqabaaabaWdbiaadUgacqGH 9aqpcaaIXaaapaqaa8qacaWGPbaajmaycqGHris5aKaaGjaadEgaca GGOaGaamiEaOWdamaaBaaajeaybaWdbiaadUgaa8aabeaajaaypeGa aiykaiabgkHiTOWaaabCaKaaG9aabaWdbiaadEhak8aadaWgaaqcba waa8qacaWGRbaapaqabaaabaWdbiaadUgacqGH9aqpcaaIXaaapaqa a8qacaWGZbaajmaycqGHris5aKaaGjaadEgacaGGOaGaamiEaOWdam aaBaaajeaybaWdbiaadUgaa8aabeaajaaypeGaaiykaaGaay5bSlaa wIa7aiaacYcacaaMe8UaaGjbVlaaysW7caWGPbGaeyOpa4Jaam4Cai aac6caaaa@68A5@ (24)

The Gauss-Legendre quadrature was used to compute the functions 16, 17, 19 and 20. However, these functions depend on integrals over infinite intervals. The integral over an infinite range must be changed into an integral over [-1,1] by using the Equations 25, 26and 27,

af(x)dx=11f(a+1+t1t)2(1t)2dt,MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWdXbqcaa0daeaapeGaamOzaaqcba0daeaapeGaamyyaaWdaeaa peGaeyOhIukajmaqcqGHRiI8aKaaajaacIcacaWG4bGaaiykaiaads gacaWG4bGaeyypa0JcdaWdXbqcaa0daeaapeGaamOzaaqcba0daeaa peGaeyOeI0IaaGymaaWdaeaapeGaaGymaaqcdaKaey4kIipakmaabm aajaaqpaqaa8qacaWGHbGaey4kaSIcdaWcaaqcaa0daeaapeGaaGym aiabgUcaRiaadshaa8aabaWdbiaaigdacqGHsislcaWG0baaaaGaay jkaiaawMcaaOWaaSaaaKaaa9aabaWdbiaaikdaa8aabaWdbiaacIca caaIXaGaeyOeI0IaamiDaiaacMcak8aadaahaaqcbauabeaapeGaaG OmaaaaaaqcaaKaamizaiaadshacaGGSaaaaa@5C73@ (25)

bf(x)dx=11f(b+1+tt1)2(t1)2dt,MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWdXbqcaa0daeaapeGaamOzaaqcba0daeaapeGaeyOeI0IaeyOh Iukapaqaa8qacaWGIbaajmaqcqGHRiI8aKaaajaacIcacaWG4bGaai ykaiaadsgacaWG4bGaeyypa0JcdaWdXbqcaa0daeaapeGaamOzaaqc ba0daeaapeGaeyOeI0IaaGymaaWdaeaapeGaaGymaaqcdaKaey4kIi pakmaabmaajaaqpaqaa8qacaWGIbGaey4kaSIcdaWcaaqcaa0daeaa peGaaGymaiabgUcaRiaadshaa8aabaWdbiaadshacqGHsislcaaIXa aaaaGaayjkaiaawMcaaOWaaSaaaKaaa9aabaWdbiaaikdaa8aabaWd biaacIcacaWG0bGaeyOeI0IaaGymaiaacMcak8aadaahaaqcbauabe aapeGaaGOmaaaaaaqcaaKaamizaiaadshacaGGSaaaaa@5D62@ (26)

abf(x)dx=ba211f((ba)2t+(b+a)2)dt.MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWdXbqcaa0daeaapeGaamOzaaqcba0daeaapeGaamyyaaWdaeaa peGaamOyaaqcdaKaey4kIipajaaqcaGGOaGaamiEaiaacMcacaWGKb GaamiEaiabg2da9OWaaSaaaKaaa9aabaWdbiaadkgacqGHsislcaWG Hbaapaqaa8qacaaIYaaaaOWaa8qCaKaaa9aabaWdbiaadAgaaKqaa9 aabaWdbiabgkHiTiaaigdaa8aabaWdbiaaigdaaKWaajabgUIiYdGc daqadaqcaa0daeaak8qadaWcaaqcaa0daeaapeGaaiikaiaadkgacq GHsislcaWGHbGaaiykaaWdaeaapeGaaGOmaaaacaWG0bGaey4kaSIc daWcaaqcaa0daeaapeGaaiikaiaadkgacqGHRaWkcaWGHbGaaiykaa WdaeaapeGaaGOmaaaaaiaawIcacaGLPaaacaWGKbGaamiDaiaac6ca aaa@5E21@ (27)

Therefore, the integrals 25, 26 and 27 were computed by applying the Gauss-Legendre quadrature rule in these transformed variables by Equations 28, 29 and 30.

af(x)dxi=1swi2(1xi)2g(a+1+xi1xi),MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWdXbqcaa0daeaapeGaamOzaaqcba0daeaapeGaamyyaaWdaeaa peGaeyOhIukajmaqcqGHRiI8aKaaajaacIcacaWG4bGaaiykaiaads gacaWG4bGaeyisISRcdaaeWbqcaa0daeaapeGaam4DaOWdamaaBaaa jeaqbaWdbiaadMgaa8aabeaaaeaapeGaamyAaiabg2da9iaaigdaa8 aabaWdbiaadohaaKWaajabggHiLdGcdaWcaaqcaa0daeaapeGaaGOm aaWdaeaapeGaaiikaiaaigdacqGHsislcaWG4bGcpaWaaSbaaKqaaf aapeGaamyAaaWdaeqaaKaaa9qacaGGPaGcpaWaaWbaaKqaafqabaWd biaaikdaaaaaaKaaajaadEgakmaabmaajaaqpaqaa8qacaWGHbGaey 4kaSIcdaWcaaqcaa0daeaapeGaaGymaiabgUcaRiaadIhak8aadaWg aaqcbauaa8qacaWGPbaapaqabaaajaaqbaWdbiaaigdacqGHsislca WG4bGcpaWaaSbaaKqaafaapeGaamyAaaWdaeqaaaaaaKaaa9qacaGL OaGaayzkaaGaaiilaaaa@6439@ (28)

bf(x)dxi=1swi2(xi1)2g(b+1+xixi1),MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWdXbqcaa0daeaapeGaamOzaaqcba0daeaapeGaeyOeI0IaeyOh Iukapaqaa8qacaWGIbaajmaqcqGHRiI8aKaaajaacIcacaWG4bGaai ykaiaadsgacaWG4bGaeyisISRcdaaeWbqcaa0daeaapeGaam4DaOWd amaaBaaajeaqbaWdbiaadMgaa8aabeaaaeaapeGaamyAaiabg2da9i aaigdaa8aabaWdbiaadohaaKWaajabggHiLdGcdaWcaaqcaa0daeaa peGaaGOmaaWdaeaapeGaaiikaiaadIhak8aadaWgaaqcbauaa8qaca WGPbaapaqabaqcaa0dbiabgkHiTiaaigdacaGGPaGcpaWaaWbaaKqa afqabaWdbiaaikdaaaaaaKaaajaadEgakmaabmaajaaqpaqaa8qaca WGIbGaey4kaSIcdaWcaaqcaa0daeaapeGaaGymaiabgUcaRiaadIha k8aadaWgaaqcbauaa8qacaWGPbaapaqabaaajaaqbaWdbiaadIhak8 aadaWgaaqcbauaa8qacaWGPbaapaqabaqcaa0dbiabgkHiTiaaigda aaaacaGLOaGaayzkaaGaaiilaaaa@6528@ (29)

abf(x) dxba2i=1swig(ba2xi+a+b2).MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWdXaqcaa0daeaapeGaamOzaaqcba0daeaapeGaamyyaaWdaeaa peGaamOyaaqcdaKaey4kIipajaaqcaGGOaGaamiEaiaacMcacaaMc8 UaamizaiaadIhacqGHijYUkmaalaaajaaqpaqaa8qacaWGIbGaeyOe I0IaamyyaaWdaeaapeGaaGOmaaaakmaaqahajaaqpaqaa8qacaWG3b GcpaWaaSbaaKqaafaapeGaamyAaaWdaeqaaaqaa8qacaWGPbGaeyyp a0JaaGymaaWdaeaapeGaam4CaaqcdaKaeyyeIuoajaaqcaWGNbGcda qadaqcaa0daeaak8qadaWcaaqcaa0daeaapeGaamOyaiabgkHiTiaa dggaa8aabaWdbiaaikdaaaGaamiEaOWdamaaBaaajeaqbaWdbiaadM gaa8aabeaajaaqpeGaey4kaSIcdaWcaaqcaa0daeaapeGaamyyaiab gUcaRiaadkgaa8aabaWdbiaaikdaaaaacaGLOaGaayzkaaGaaiOlaa aa@610C@ (30)

Details of how to compute the nodes and weights to apply the Gauss-Legendre quadrature can be found in Hildebrand (1974).

An R package (R Development Core Team, 2017) denoted by SMR (Batista; Ferreira, 2012) was developed to apply the Gauss-Legendre quadratures to compute the cumulative distribution functions 17 and 20 and the probability density functions 16 and 19. The numerical transformations given by Equations 28, 29 and 30 and the methods of computation of the nodes and weights of Gauss-Legendre quadrature, as described in Hildebrand (1974) and Gil, Segura and Temme (2007) were used in the R codes of th e implemented package. The quantiles were computed using the Newton-Raphson method solving equations formed by equating the functions 17 or 20 to p, where 0 < p < 1 is the cumulative probability (Gil, Segura, Temme, 2007), which is known. These computations make use of numerical quadratures of the respectively probability density functions 16 or 19, which are the first derivatives of 17 or 20.

Besides computing the cumulative distribution, probability density and quantile functions, the SMR package generates random samples of size N by Monte Carlo simulation. For this, random samples of size n, X 1, X 2, …, X n , are generated, where the X i s are independent and identically distributed standard normal variables, N(0, 1), for i = 1, 2, …, n. A random variable U, distributed as a chi-square variable, χν2MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHhpWydaqhaaWcbaGaeqyVd4gabaGaaGOmaaaaaaa@3A6E@ , is simulated, which is independently distributed of the X i ’s, where v > 0 is the degrees of freedom. Finally, the following transformation is performed

Q¯=[max(Xi)+min(Xi)]/2Uν.MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaeeaaaaaa aaa8qaceWGrbGbaebacqGH9aqpkmaalaaajaaqpaqaa8qacaGGBbGa amyBaiaadggacaWG4bGaaiikaiaadIfak8aadaWgaaqcbauaa8qaca WGPbaapaqabaqcaa0dbiaacMcacqGHRaWkcaWGTbGaamyAaiaad6ga caGGOaGaamiwaOWdamaaBaaajeaqbaWdbiaadMgaa8aabeaajaaqpe Gaaiykaiaac2facaGGVaGaaGOmaaWdaeaak8qadaGcaaqcaa0daeaa k8qadaWcaaqcaa0daeaapeGaamyvaaWdaeaapeGaeqyVd4gaaaqcba uabaaaaKaaajaac6caaaa@4FE7@

With infinity degrees of freedom v = ∞, it is suffice to compute

W¯=max(Xi)+min(Xi)2.MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaeeaaaaaa aaa8qaceWGxbGbaebacqGH9aqpkmaalaaajaaqpaqaa8qacaWGTbGa amyyaiaadIhacaGGOaGaamiwaOWdamaaBaaajeaqbaWdbiaadMgaa8 aabeaajaaqpeGaaiykaiabgUcaRiaad2gacaWGPbGaamOBaiaacIca caWGybGcpaWaaSbaaKqaafaapeGaamyAaaWdaeqaaKaaa9qacaGGPa aapaqaa8qacaaIYaaaaiaac6caaaa@4932@

The process was repeated N times and the required sample values of Q¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGrbWdayaaraaaaa@3714@ or of W¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaraaaaa@371A@ were obtained. The package SMR provides the following functions, where np is the number of nodes and weights of the Gauss-Legendre quadrature:

dSMR(x, n, nu, np=32): computes values of the probability density function, given in (16) or (19);

pSMR(x, n, nu, np=32): computes values of the cumulative distribution function, given in (17) or (20);

qSMR(p, n, nu, np=32): computes quantiles of the externally studentized normal midrange;

rSMR(N, n, nu=Inf): drawn random samples of the externally studentized normal midrange.

The user can choose the argument nu as finity or infinity value. If nu=Inf, values of the probability density, cumulative distribution and quantile functions of the normal midrange (standard normal midrange) are computed. If the argument nu is not specified in the rSMR function, the default value Inf is used and random samples from the normal midrange distribution are drawn.

Performance

Evaluations of the accuracy of quadratures for computing the cumulative distribution functions 17 and 20 and for computing quantiles were performed. The SMR package was used for this purpose. There are no cumulative probabilities or quantiles of the externally studentized normal midrange to be compared, since reports of this distribution were not found in the literature. Two strategies were proposed to verify the accuracy. First, Monte Carlo simulations were used to obtain quantiles and cumulative probabilities and to compare them to those obtained by using the Gauss-Legendre quadratures from the SMR package. Second, two different number of quadrature points were used to compute these quantities. Thus, the quadrature errors were computed by comparing these two values, as showed in 24.

Validations of the algorithm by Monte Carlo simulation were done using the R software with the SMR package (Batista; Ferreira, 2012). A random sample of externally studentized normal midrange, Q¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGrbWdayaaraaaaa@3714@ , of size N = 1,000,001 was simulated using the rSMR function of the SMR package, following the procedure described above. If the degrees of freedom were ∞, then this function generates random sample from the standard normal midrange, W¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGxbWdayaaraaaaa@371A@ .

Therefore, given a quantile q¯MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGXbGbaebaaaa@3724@ or w¯MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWG3bGbaebaaaa@372A@ , the cumulative probability is computed, respectively, by

P(Q¯q¯)=i=1NI(Q¯iq¯)N, ν< orMathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaeeaaaaaa aaa8qacaWGqbGaaiikaiqadgfagaqeaiabgsMiJkqadghagaqeaiaa cMcacqGH9aqpkmaalaaajaaqpaqaaOWdbmaaqahajaaqpaqaa8qaca WGjbaajeaqpaqaa8qacaWGPbGaeyypa0JaaGymaaWdaeaapeGaamOt aaqcdaKaeyyeIuoajaaqcaGGOaGabmyuayaaraGcpaWaaSbaaKqaaf aapeGaamyAaaWdaeqaaKaaa9qacqGHKjYOceWGXbGbaebacaGGPaaa paqaa8qacaWGobaaaiaacYcacaaMe8UaeqyVd4MaeyipaWJaeyOhIu QaaGjbVlaaysW7caqGVbGaaeOCaaaa@5920@

P(W¯w¯)=i=1NI(W¯iw¯)N, ν=.MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaeeaaaaaa aaa8qacaWGqbGaaiikaiqadEfagaqeaiabgsMiJkqadEhagaqeaiaa cMcacqGH9aqpkmaalaaajaaqpaqaaOWdbmaaqahajaaqpaqaa8qaca WGjbaajeaqpaqaa8qacaWGPbGaeyypa0JaaGymaaWdaeaapeGaamOt aaqcdaKaeyyeIuoajaaqcaGGOaGabm4vayaaraGcpaWaaSbaaKqaaf aapeGaamyAaaWdaeqaaKaaa9qacqGHKjYOceWG3bGbaebacaGGPaaa paqaa8qacaWGobaaaiaacYcacaaMe8UaeqyVd4Maeyypa0JaeyOhIu QaaiOlaaaa@54EB@

Table 1 shows the values of the distribution function computed by 64 Gauss-Legendre quadrature points and by Monte Carlo simulations (with N=1,000,001 observations in the Monte Carlo sample). Several combinations of n and v were used for some particular choices of the quantiles q¯MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGXbGbaebaaaa@3724@ or w¯MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWG3bGbaebaaaa@372A@ . The cumulative probabilities computed by Gauss-Legendre quadrature were close to those obtained by Monte Carlo simulations (MC). The MC error is proportional to 1/NMathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIXaGaai4lamaakaaapaqaa8qacaWGobaaleqaaaaa@3891@ (Ciftja and Wexler, 2003), which in this case is 0.001. The two methods showed differences in the third and fourth decimal places (Table 1), as expected. These results show, in principle, that the Gauss-Legendre quadrature has at least three significant digits of accuracy.

Table 1: Computed values of

P(Q¯q¯)MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaacI caceWGrbGbaebacqGHKjYOceWGXbGbaebacaGGPaaaaa@3BD6@

, using the Gauss-Legendre Quadrature and by simulation using Monte Carlo method, with sample size n, and v degrees of freedom, considering N=1,000,001 simulations. 

Values Method
n v or Quadrature MC
15 4 4.000 0.999712051088144 0.999719000280999
30 7 2.000 0.999521701960583 0.999520000479999
45 10 4.000 0.999999689708233 0.999999999999999
60 25 1.000 0.996635357142795 0.996665003334996
20 2 1.000 0.941476242577670 0.941947058052942
30 5 0.300 0.786876942543113 0.786846213153786
90 40 0.200 0.748082418017427 0.748314251685748
30 10 0.000 0.500000000000000 0.500382499617500
40 5 -1.000 0.016709246604515 0.016733983266017
20 20 -0.400 0.147628604257637 0.147586852413147
15 0.300 0.778662390742543 0.778745221254779
60 0.800 0.991576069342297 0.991477008522991

The quadrature error ɛ can also be computed increasing the number of points and calculating the difference between the two results. Cumulative probabilities computed by using s=64 and i=250 quadrature points and the errors obtained by calculating the differences between these values are shown in Table 2. The maximum observed error is of the order of 10-10, showing that with 64 quadrature points a high precision was achieved, for computing 17 or 20. Several other combinations of n, v and q¯MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGXbGbaebaaaa@3724@ or w¯MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWG3bGbaebaaaa@372A@ were used to compute the Monte Carlo errors and their maximum value is still the same.

Table 2: Computed values of

P(Q¯q¯)MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaacI caceWGrbGbaebacqGHKjYOceWGXbGbaebacaGGPaaaaa@3BD6@

, using the Gauss-Legendre Quadrature for s=64 and i=250 points of quadrature, with sample size n, and v degrees of freedom. 

Values Quadrature
n v or s = 64 points i = 250 points ɛ
15 4 4.000 0.999712051088145 0.99971205062441360 4.637313 x 10-10
30 7 2.000 0.999521701960583 0.9995217019605828 1.110223 x 10-10
45 10 4.000 0.999999689708230 0.99999968970816311 6.694644 x 10-10
60 25 1.000 0.996635357142795 0.99663535714279172 3.330669 x 10-10
20 2 1.000 0.941476242577670 0.94147624257766860 1.443290 x 10-10
30 5 0.300 0.786876942543113 0.78687694254311258 4.440892 x 10-10
90 40 0.200 0.748082418017427 0.74808241801730846 1.185718 x 10-10
30 10 0.000 0.500000000000002 0.50000000000000200 1.000000 x 10-10
40 5 -1.000 0.016709246604515 0.01670924660451461 3.920475 x 10-10
20 20 -0.400 0.147628604257637 0.14762860425763724 2.498002 x 10-10
15 0.300 0.778662390742543 0.77866239074254306 3.330669 x 10-10
60 0.800 0.991576069342296 0.99157606934229103 5.662137 x 10-10

Table 3 shows quantiles for a settled value of the cumulative probabilities of 0.95, using the qSMR function of SMR package (Batista; Ferreira, 2012, 2014). In this case, almost all the quantiles were computed using 64 quadrature points, except, for the case of v = 1, where this number of points was insufficient. In such circumstances, there are two alternatives: a) refine the quadrature, dividing the integral interval into small subintervals and approximate the integral by a sum of computation on each subinterval; or b) increasing the number of quadrature points. In this case, we opted for the latter, using 250 quadrature points. The results were shown using 3 decimal places, for v degrees of freedom, of 1(1)20(5)30, 50(50)200, 300, 1,000 and ∞, and for sample sizes n of 2(1)10(5)50(25)100. When v → ∞, thus s 2 → σ2 and the studentized midrange cumulative distribution function from standard normal populations F(q¯;n,ν)MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGgbGaaiikaiqadghagaqeaiaacUdacaWGUbGaaiilaiabe27a UjaacMcaaaa@3D62@ , given in 20, tends to FW¯(w¯;n)MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGgbWdamaaBaaaleaapeGabm4vayaaraaapaqabaGcpeGaaiik aiqadEhagaqeaiaacUdacaWGUbGaaiykaaaa@3C68@ , the standardized midrange cumulative distribution function, given in 17. If the desired value of q¯MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWGXbGbaebaaaa@3724@ can not be found in Table 3, the SMR package can be used or linear interpolations can be applied to compute this quantile.

Table 3:Upper quantile of the distribution of externally studentized normal midrange

(q¯ or w¯)MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiqadg hagaqeaiaabccacaWGVbGaamOCaiaabccaceWG3bGbaebacaGGPaaa aa@3CA3@

for different degrees of freedom v and sample sizes (n), according the following probability event:

P(Q¯q¯)MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaacI caceWGrbGbaebacqGHKjYOceWGXbGbaebacaGGPaaaaa@3BD6@

= 0.95 or

P(W¯w¯)MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaacI caceWGxbGbaebacqGHKjYOceWG3bGbaebacaGGPaaaaa@3BE2@

= 0.95. 

Sample size (n)
v 2 3 4 5 6 7 8 9 10
1 4.464 3.799 3.445 3.219 3.060 2.941 2.846 2.770 2.705
2 2.065 1.757 1.594 1.490 1.417 1.362 1.319 1.284 1.254
3 1.664 1.416 1.285 1.201 1.143 1.098 1.064 1.035 1.012
4 1.507 1.283 1.164 1.088 1.035 0.995 0.964 0.938 0.917
5 1.425 1.213 1.100 1.029 0.978 0.941 0.911 0.887 0.867
6 1.374 1.169 1.061 0.992 0.944 0.907 0.879 0.855 0.836
7 1.340 1.140 1.034 0.967 0.920 0.885 0.857 0.934 0.815
8 1.315 1.119 1.015 0.949 0.903 0.868 0.841 0.819 0.800
9 1.296 1.103 1.001 0.936 0.890 0.856 0.829 0.807 0.789
10 1.282 1.091 0.990 0.925 0.880 0.846 0.820 0.798 0.780
15 1.240 1.055 0.957 0.895 0.851 0.819 0.793 0.772 0.754
16 1.235 1.051 0.953 0.891 0.848 0.815 0.789 0.769 0.751
17 1.230 1.047 0.950 0.888 0.845 0.812 0.787 0.766 0.748
18 1.226 1.044 0.947 0.885 0.842 0.810 0.784 0.763 0.746
19 1.223 1.041 0.944 0.883 0.840 0.807 0.782 0.761 0.744
20 1.220 1.038 0.942 0.881 0.838 0.805 0.780 0.759 0.742
25 1.208 1.028 0.933 0.872 0.830 0.798 0.772 0.752 0.735
30 1.200 1.021 0.927 0.867 0.824 0.793 0.768 0.747 0.730
50 1.185 1.009 0.915 0.856 0.814 0.783 0.758 0.738 0.721
100 1.174 0.999 0.906 0.848 0.806 0.775 0.751 0.731 0.714
150 1.170 0996 0.904 0.845 0.804 0.773 0.748 0.729 0.712
200 1.168 0.994 0.902 0.844 0.803 0.772 0.747 0.727 0.711
300 1.167 0.993 0.901 0.842 0.801 0.770 0.746 0.726 0.710
1000 1.164 0.991 0.899 0.841 0.800 0.769 0.744 0.725 0.708
1.163 0.990 0.898 0.840 0.799 0.768 0.744 0.724 0.708
Sample size (n)
v 15 20 25 30 35 40 45 50 75 100
1 2.492 2.366 2.280 2.215 2.165 2.124 2.090 2.060 1.957 1.893
2 1.156 1.099 1.059 1.030 1.007 0.988 0.972 0.959 0.912 0.882
3 0.933 0.887 0.855 0.832 0.813 0.798 0.785 0.774 0.736 0.713
4 0.846 0.804 0.775 0.754 0.737 0.723 0.712 0.702 0.668 0.646
5 0.800 0.760 0.733 0.713 0.697 0.684 0.673 0.664 0.632 0.611
6 0.771 0.733 0.707 0.687 0.672 0.660 0.649 0.640 0.609 0.590
7 0.752 0.715 0.689 0.670 0.656 0.643 0.633 0.625 0.594 0.575
8 0.738 0.702 0.677 0.658 0.643 0.632 0.622 0.613 0.583 0.564
9 0.728 0.692 0.667 0.649 0.634 0.623 0.613 0.604 0.575 0.556
10 0.719 0.684 0.660 0.641 0.627 0.616 0.606 0.598 0.568 0.550
15 0.696 0.661 0.638 0.620 0.607 0.596 0.586 0.578 0.550 0.532
16 0.693 0.659 0.635 0.618 0.604 0.593 0.584 0.576 0.548 0.530
17 0.691 0.656 0.633 0.616 0.602 0.591 0.582 0.574 0.546 0.528
18 0.688 0.645 0.631 0.614 0.600 0.589 0.580 0.572 0.544 0.527
19 0.686 0.652 0.629 0.612 0.598 0.587 0.578 0.570 0.542 0.525
20 0.685 0.651 0.628 0.610 0.597 0.586 0.577 0.569 0.541 0.524
25 0.678 0.645 0.622 0.605 0.591 0.580 0.571 0.563 0.536 0.519
30 0.674 0.640 0.618 0.601 0.587 0.577 0.568 0.560 0.532 0.515
50 0.665 0.632 0.610 0.593 0.580 0.569 0.560 0.553 0.526 0.509
100 0.659 0.626 0.604 0.588 0.575 0.564 0.555 0.548 0.521 0.504
150 0.657 0.625 0.602 0.586 0.573 0.562 0.553 0.546 0.519 0.503
200 0.656 0.624 0.601 0.585 0.572 0.561 0.553 0.545 0.518 0.502
300 0.655 0.623 0.601 0.584 0.571 0.561 0.552 0.544 0.518 0.501
1000 0.654 0.621 0.599 0.583 0.570 0.559 0.551 0.543 0.517 0.500
0.653 0.621 0.599 0.582 0.569 0.559 0.550 0.543 0.516 0.500

In Table 3, if the number of degrees of freedom and the cumulative probability is settled, the quantiles decrease as the sample size increases. Comparing these results of the studentized normal midrange with quantiles of the studentized normal range in the same circumstances (Newman, 1939), this is not observed. Fixing the number of degrees of freedom and the probability, the studentized normal range quantiles increase as the sample size increases. This difference can be observed in Figure 1, where the studentized range and midrange probability density functions were plotted, for v = 10 with n = 10 and n = 1,000. Another interesting observation is that fixing the sample size and the cumulative probability, the studentized normal midrange quantiles decrease as the number of degrees of freedom increases (Table 3). The same can be observed for the studentized normal range quantiles. In Figure 1(a), for n=10 and n=1,000, with v = 10, the quantiles for P(Qq) = F(q;n,v) = 0.95 are q n=10 = 5.6 and q n=1,000 = 10.5, respectively. Increasing the sample size there is a translation of the range probability density function to the right. This occurs because the range increases as sample size increases.

Figure 1: Probability density function (pdf) of the externally studentized normal range (a) and midrange (b), considering v = 10 with n = 10 and n = 1,000. 

In Figure 1(b), for n=10 and n=1,000, with v = 10, the quantiles for P(Q¯q¯)MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGqbGaaiikaiqadgfagaqeaiabgsMiJkqadghagaqeaiaacMca aaa@3BF5@ are q n=10 = 0.78 and q n=1,000 = 0.45, respectively. When the sample size increases there is an increased concentration of data around its center without change its central position. This concentration of data decreases the value of the quantile in larger sample, considering the same cumulative probability.

Applications

The development of externally studentized normal midrange distribution allows theat several applications can be performed. One of the already results of this distribution is the midrangeMCP package that performs four multiple comparison tests based on this distribution (Batista; Ferreira, 2014). The tests developed and implemented in this package are versions similar to the Tukey, SNK and Scott-Knott tests. The first two, in their original version, are based on the distribution of externally studentized normal range. The Scott-Knott test is based on the likelihood ratio as a criterion for separating groups of means. However, in the midrangeMCP package, these tests were based on the distribution of externally studentized normal midrange, which present high performance when evaluated for the type I error rates and power, although results are in the publishing process. Another interesting application of this distribution is in the building of control charts in the statistical area of quality control, replacing the normal standardized range or the studentized range.

CONCLUSIONS

The analytical equations of the distribution of the externally studentized normal midrange ( Q¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyuayaara aaaa@36E5@ ), as well as of the distribution of normal midrange ( W¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4vayaara aaaa@36EB@ ), were achieved. Probability density, cumulative distribution and quantiles functions were obtained computationally for Q¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyuayaara aaaa@36E5@ and W¯MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4vayaara aaaa@36EB@ . The algorithms were proposed and implemented using Gauss-Legendre quadrature and the Newton-Raphson method in R software, resulting in the SMR package, available for download in the CRAN site. The implemented routines showed high accuracy proved by using Monte Carlo simulations and by comparing results with different number of quadrature points. Regarding to the precision to obtain the quantiles for cases where the degrees of freedom are close to 1 and the percentiles are close to 100%, it is recommended to use more than 64 quadrature points.

ACKNOWLEDGMENTS

We would like to thank CNPq and CAPES for their financial support.

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Received: December 15, 2016; Accepted: February 15, 2017

*Corresponding author: ben.deivide@gmail.com

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