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Pesquisa Operacional

Print version ISSN 0101-7438On-line version ISSN 1678-5142

Pesqui. Oper. vol.38 no.3 Rio de Janeiro Sept./Dec. 2018

http://dx.doi.org/10.1590/0101-7438.2018.038.03.0555 

Articles

THE UNIT-LOGISTIC DISTRIBUTION: DIFFERENT METHODS OF ESTIMATION

André Felipe Berdusco Menezes1  * 

Josmar Mazucheli1 

Sanku Dey2 

1Departamento de Estatística, Universidade Estadual de Maringá, 87020-900 Maringá, PR, Brasil. E-mails: andrefelipemaringa@gmail.com; jmazucheli@gmail.com

2Department of Statistics, St. Anthony’s College, Shillong, Meghalaya, India. E-mail: sankud66@gmail.com

ABSTRACT

This paper addresses the different methods of estimation of the unknown parameters of a two-parameter unit-logistic distribution from the frequentist point of view. We briefly describe different approaches, namely, maximum likelihood estimators, percentile based estimators, least squares estimators, maximum product of spacings estimators, methods of minimum distances: Cramér-von Mises, AndersonDarling and four variants of Anderson-Darling. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation for both small and large samples. The performances of the estimators have been compared in terms of their relative bias, root mean squared error, average absolute difference between the theoretical and empirical estimate of the distribution functions and the maximum absolute difference between the theoretical and empirical distribution functions using simulated samples. Also, for each method of estimation, we consider the interval estimation using the Bootstrap confidence interval and calculate the coverage probability and the average width of the Bootstrap confidence intervals. Finally, two real data sets have been analyzed for illustrative purposes.

Keywords: Unit-Logistic distribution; Monte Carlo simulations; estimation methods; parametric Bootstrap

1 INTRODUCTION

Tadikamalla & Johnson 30 introduced a new probability distribution with support on (0, 1) and named the distribution as L B distribution by using transformations of logistic variables. They obtained the distribution as follows:

X=g-1 Y-γδ, (1)

where Y is a standard Logistic distribution, g(·) is some suitable function and γ ∈ ℝ, δ> 0 are parameters. The choice of g(·) determines the support of the distribution, hence from 30, by taking:

gX=logX1-X, (2)

we can obtain the L B distribution, hereafter we refer it as unit-Logistic distribution, with probability density function (PDF) given by:

f(x | γ, δ) =δ eγ xδ-1 (1x)δ-1xδ eγ+(1x)δ2, 0<x<1. (3)

In spite of its versatility, this distribution did not receive much attention in the literature. Nevertheless, recently, the basic properties and a regression analysis was studied by 5. The authors introduced an alternative parametrization, where one parameter is the median. Following this parametrization, they defined the PDF as:

f(x | μ, β) =β μβ xβ-1 (1μ)β (1x)β-1(1μ)β xβ+μβ (1x)β2, 0<x<1 (4)

where 0 < μ < 1 is the median of X and β > 0 is the shape parameter. The corresponding cumulative distribution function and quantile function are written respectively as:

F(x | μ, β) =1+μ (1x)x (1μ)β-1, 0<x<1 (5)

and

Q(p | μ, β) =μ p1/β(1μ) (1p)1/β+μ p1/β, 0<p<1. (6)

Note that when we set μ = 0.5 and β = 1 in (4), the PDF of the unit-logistic distribution simply becomes the PDF of the standard uniform distribution. As we can see in Figure 1 the unitLogistic density is uni-modal (or uni-antimodal), increasing, decreasing, or constant, depending on the values of the parameters.

Figure 1 Behavior of the probability density function of unit-Logistic distribution for some values of μ and β

The objective of this paper is to introduce different methods of estimation for the unknown parameters that index the unit-Logistic distribution and to study the behavior of these estimators for different sample sizes and for different parameter values. In particular, we compare the maximum likelihood estimators, maximum product of spacings estimators, estimators based on percentiles, least-squares estimators, weighted least-squares estimators, Cramér-von Mises estimators and Anderson-Darling estimators and four of its variants. Since, it is difficult to compare theoretically the performances of the different estimators, we perform extensive simulations to compare the performances of the different estimation methods based on relative bias, root mean squared error, average absolute difference between the theoretical and empirical estimate of the distribution functions, and maximum absolute difference between the theoretical and empirical distribution functions. Also, for each method of estimation, we

consider the interval estimation using the Bootstrap confidence interval 12 and calculate the coverage probability and the average width of the confidence interval.

The uniqueness of this study comes from the fact that thus far, no attempt has been made to compare all these estimators for the two-parameter unit-logistic distribution. Comprehensive comparisons of estimation methods for other distributions have been performed in the literature: see 13

for generalized Exponential distribution, 17 for generalized Rayleigh distributions, 31 for Weibull distribution, 22 for weighted Lindley distribution, 10 for Marshall-Olkin extended Lindley distribution, 7 for weighted Exponential distribution, 21 for Marshall-Olkin extended Exponential distribution, 9 for the Kumaraswamy distribution, 8 for the Exponentiated Chen distribution, 27 for the Poisson-exponential distribution, 24 for the Alpha logarithmic transformed Weibull distribution and 23 for the power inverse Lindley distribution.

The final motivation of the paper is to show how different frequentist estimators of this distribution perform for different sample sizes and different parameter values and to develop a guideline for choosing the best estimation method for the unit-logistic distribution, which we think would be of interest to applied statisticians.

The paper is organized as follows. In Section 2 we discuss the eleven estimation methods considered in this paper. The performance of the proposed estimation procedures is studied through a Monte Carlo simulation and is presented in Section 3. In section 4, the methodology developed in this manuscript and the usefulness of the unit-Logistic distribution is illustrated by using two real data examples. Some concluding remarks are presented in Section 5.

2 METHODS OF ESTIMATION

In this section, we describe seven methods and four variants of AD test for estimating the parameters, μ and β, associated to the unit-Logistic distribution. For all methods, it is assumed that x = (x 1 , ... , x n ) is a random sample of size n from the unit-Logistic distribution with PDF given by (4) and unknown parameters μ and β. Also, let x(1) < ... < x(n) be the corresponding order sample statistics.

2.1 Method of Maximum Likelihood

Undoubtedly the method of maximum likelihood is the most popular method in statistical inference, mainly because of its many appealing properties. For instance, the maximum likelihood estimates are asymptotically unbiased, efficient, consistent, invariant under parameter transformation and asymptotically normally distributed (see, e.g., 18), (25), (28).

The log-likelihood function of unit-Logistic distribution based on the random sample x = (x 1, ..., x n ) can be written as:

l=(μ, β | x)=n log β+n β log (1μ)+n β log μ +(β1) i=1n log xi+(β1) i=1n log (1xi) -2 i=1n log (1μ)β xiβ+μβ (1xi)β. (7)

The maximum likelihood estimates μ^MLE and β^MLE of the parameters μ and β, respectively, can be obtained by maximizing (7), or equivalently solving the following normal equations:

lβ=n β-μ βμ 1-μ-2 β i=1n (1μ)β-1 xiβ-μβ-1 1-xiβ(1μ)β xiβ+μβ 1-xiβ=0, (8)

lβ=nβ+n log(1μ)+n log μ+i=1n log xi+i=1n log(1xi)-2 i=1nxiβ log (1μ)+log xi (1μ) β+2 μβ (1xi)β log (1xi)+log μ(1μ)β xiβ+μβ (1xi)β=0. (9)

Confidence intervals can be obtained by using the large sample distribution of the MLEs, which is normally distributed with the covariance matrix given by the inverse of the Fisher information since regularity conditions are satisfied.

2.2 Method of Maximum Product of Spacings

The maximum product of spacings (MPS) method was introduced by 2), (3 as an alternative to MLE for estimating parameters of continuous univariate distributions. Ranneby 26 independently derived the same method as an approximation to the Kullback-Leibler measure of information.

The uniform spacings of a random sample from unit-Logistic distribution is defined as:

Di(μ, β)=F(x(i) | μ, β)F(x(i1) | μ, β), i=1,..., n,F(x(0) | μ, β)=0 and F(x(n+1) | μ, β) = 0.

Clearly, D 0(μ, β) + D 1(μ, β) + ... + D n +1 (μ, β) = 1.

From 2), (3, the MPSEs, μ^MPS and β^MPS are the values of μ and β, which maximize the geometric mean of the spacings:

G(μ, β | x)=i=1n+1 Diμ, β1n+1 (10)

or, equivalently, by maximize its logarithm:

H(μ, β | x)=1n+1 i=1n+1 log Di. (11)

The estimators μ^MPS and β^MPS of the parameters α and β can also be obtained by solving the nonlinear equations:

μ H(μ, β) = 1n+1 i=1n+1 1Di(μ, β) Δ1(x(i) | μ, β)Δ1(x(i1) | μ, β) =0,β H(μ, β) = 1n+1 i=1n+1 1Di(μ, β) Δ2(x(i) | μ, β)Δ2(x(i1) | μ, β)=0,

where

Δ1 (xi | μ, β) =μβ-1 β xi-βμ1x(i)1β(μ1) x(i) μ-β μ1x(i)1β+12 (12)

and

Δ2 (xi:n | μ, β) =log μ+log(1x(i))log x(i)log(1μ) x(i)β μβμ1x(i)1β(μ1) x(i) μ-β μ1x(i)1β+12. (13)

It is noteworthy that the MPSE is as efficient as ML estimation and consistent under more general conditions than the ML estimators 3.

2.3 Method of Percentiles

If the data come from a distribution function which has a closed form, then we can estimate the unknown parameters by fitting straight line to the theoretical points obtained from the distribution function and the sample percentile points. This method was developed by 15),(16 to estimate the parameters of the Weibull distribution.

Since the unit-Logistic distribution has an explicit cumulative distribution function, (5), it is feasible to use the same concept to derive estimators for μ and β. If p i denotes some estimate of Fx (i) | μ, β , then the percentiles estimators, μ^PCE and β^PCE can be obtained by minimizing, with respect to μ and β, the nonlinear function:

P(μ, β | x)= i=1n x(i)μ pi1/β (1μ) (1pi)1/β+μ pi1/β2, (14)

where pi=in+1 is an unbiased estimator of F(x (i) | μ, β) It is to be mentioned here that there are several possible choices for p i , interested readers may refer to 20.

2.4 Methods of Least Squares

The least square methods were originally proposed by 29 to estimate the parameters of the Beta distributions. Suppose that F(X(i)) denotes the distribution function of the order statistics from the random sample x = (x 1 , ..., x n ). An important result from the probability shows that F(X (i)) ∼ Beta(i, ni + 1). Therefore, we have:

E F X(i)=in+1 and V F X(i)=i n-i+1n+12 n+2 (15)

for further details see 14. Using the expectations and variances, we obtain two variants of the least squares methods.

2.4.1 Ordinary Least Squares

In case of unit-Logistic distribution, the ordinary least square estimates μ^OLS and β^OLS of the parameters μ and β can be obtained by minimizing the function:

S (μ, β | x)= i=1n F x(i) | μ, βin+12 (16)

with respect to μ and β. Alternatively, these estimates can also be obtained by solving the following nonlinear equations:

i=1nFxi | μ, β-in+1 Δ1 xi | μ, β=0,i=1nFxi | μ, β-in+1 Δ2 xi | μ, β=0

where Δ1(· | μ, β) and Δ2(· | μ, β) are given by Equations (12) and 13, respectively.

2.4.2 Weighted Least Squares

For the unit-Logistic distribution, the weighted least square estimates of μ and β, say μ^WLS and β^WLS , respectively are obtained by minimizing the function:

W (μ, β | x)= i=1n (n+1)2 (n+2)i (ni+1) F x(i) | μ, βin+12 (17)

with respect to μ and β. Equivalently, these estimates are the solution of the following nonlinear equations:

i=1n n+12 n+2in-i+1Fxi | μ, β-in+1 Δ1 xi | μ, β=0,i=1n n+12 n+2in-i+1Fxi | μ, β-in+1 Δ2 xi | μ, β=0

where Δ1(· | μ, β) and Δ2(· | μ, β) are defined in Equations (12) and 13, respectively.

2.5 Methods of Minimum Distances

Here, we will discuss some methods based on the test statistics of Cramár-von Mises, Anderson-Darling and four variants of the Anderson-Darling test, whose acronyms are ADR, AD2R, AD2L and AD2. Mainly, these methods determine the values of parameters that minimize the distance between the theoretical and empirical cumulative distribution functions (see for further details e.g., 6), (19). The expressions for each method are presented in Table 1.

Table 1 Expressions for the methods based on the minimum distances. 

Acronyms Expressions
CvM wn2=112n+i=1nxi-2i-12n2
AD An2=-n-1n+i=1n 2i-1log xi+log1-xn+1-i
ADR Rn2=n2-2i=1n xi-1ni=1n2i-1 log1-xn+1-i
AD2R rn2=2i=1n log1-xi+1ni=1n 2i-11-xn+1-i
AD2L ln2=2i=1n log xi+1ni=1n 2i-1xi
AD2 an2=2i=1n log xi+log(1-xi)+1ni=1n 2i-1xi+2i-11-xn+1-i

For illustrative purposes, we have presented only the expressions used for the estimation of the parameters for the Cramér-von Mises and Anderson-Darling methods.

2.5.1 Method of Cramér-von Mises

In regard to unit-Logistic distribution, the Cramér-von- Mises estimates μ^CvM and β^CvM are obtained by minimizing with respect to μ and β the function:

C(μ, β | x)=112n+ i=1n F x(i) | μ, β2i12n2. (18)

The estimates can also be obtained by solving the following nonlinear equations:

i=1n Fxi | μ, β-2i-12n Δ1 xi | μ, β=0,i=1n Fxi | μ, β-2i-12n Δ2 xi | μ, β=0

where Δ1(· | μ, β) and Δ2(· | μ, β) are specified in Equations (12) and 13, respectively.

2.5.2 Method of Anderson-Darling

1 developed a test, as an alternative to statistical tests for detecting sample distributions departure from normality. Using these test statistics, we can obtain the Anderson-Darling estimates, μ^ADE and β^ADE by minimizing the function

A(μ, β | x)=n1n i=1n (2i 1) log F x(i) | μ, β+log F¯ x(n+1i) | μ, β. (19)

with respect to μ and β. Equivalently, these estimates are the solution of the following nonlinear equations:

i=1n 2i-1 Δ1 xi | μ, βFxi | μ, β-Δ1 xn+1-i | μ, βFxn+1-i | μ, β=0,i=1n 2i-1 Δ2 xi | μ, βFxi | μ, β-Δ2 xn+1-i | μ, βFxn+1-i | μ, β=0

where Δ1(· | α, β) and Δ2(· | α, β) are given by (12) and (13), respectively.

3 SIMULATION RESULTS

In this section we conduct a Monte Carlo simulation study to compare the performance of the frequentist estimators discussed in the previous sections. The methods are compared for sample sizes n = {20, 50, 100, 200}. We generate M = 5.000 pseudo-random samples from unit-Logistic distribution using the inverse transform method with parameters μ = {0.2, 0.4, 0.6, 0.8} and β = {0.5, 1.5, 2.0}.

All simulations are done in Ox version 7.10, (see 11), using the MaxBFGS subroutine for numerical optimizations. For each estimate, we calculate the relative bias, root mean-squared error (RMSE), the average absolute difference between the theoretical and empirical estimate of the distribution functions (D abs), and the maximum absolute difference between the theoretical and empirical distribution functions (D max). These measures are obtained using the following formulae:

Bias Θ^ = 1M i=1M Θ^i-ΘΘ, (20)

RMSE Θ^ = 1M i=1M Θ^i-Θ2, (21)

Dabs=1M×n i=1Mj=1n | F yij | Θ-Fyij | Θ^|, (22)

Dmax=1M i=1M maxj | F yij | Θ-Fyij | Θ^|. (23)

where Θ = (μ, β). Due to space constraint, we report the results only for μ = (0.2, 0.8) and β = (0.5, 2). The results for other combinations are summarized by their ranks in Tables 6 and 11, however this can be obtained from the corresponding author on request.

Table 2 Simulations results for μ = 0.2 and β = 0.5. 

n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
20 Bias( β^ ) 0.0254 -0.0385 0.07110 0.0697 0.0526 0.07711 0.0709 -0.0708 -0.0093 0.0001 0.0022
RMSE( β^ ) 0.2142 0.2163 0.36110 0.36611 0.2587 0.2718 0.2265 0.2001 0.2356 0.2989 0.2234
Bias( μ^ ) 0.1083 0.1269 0.0881 0.16711 0.12910 0.1097 0.1072 0.1084 0.1096 0.1118 0.1085
RMSE( μ^ ) 0.6422 0.69310 0.6768 0.72211 0.6577 0.6456 0.6411 0.6423 0.6444 0.6819 0.6445
D^abs 0.0988 0.0972 0.09811 0.0973 0.09810 0.0989 0.0987 0.0975 0.0961 0.0976 0.0974
D^max 0.2227 0.2213 0.22311 0.2228 0.22210 0.2225 0.2229 0.2226 0.2201 0.2202 0.2224
Total 263 325 5110 5110 509 468 336 274 211 357 242
n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
50 Bias( β^ ) 0.0076 -0.04611 0.0012 -0.0011 0.0167 0.0269 0.0248 -0.04410 -0.0075 -0.0044 0.0023
RMSE( β^ ) 0.1243 0.1437 0.20110 0.20511 0.1426 0.1458 0.1242 0.1221 0.1375 0.1879 0.1274
Bias( μ^ ) 0.0466 0.05910 0.07611 0.0463 0.0559 0.0478 0.0464 0.0465 0.0477 0.0441 0.0462
RMSE( μ^ ) 0.3982 0.4439 0.46011 0.45510 0.4057 0.4006 0.3981 0.3983 0.4005 0.4328 0.3994
D^abs 0.0622 0.0626 0.06210 0.06211 0.0623 0.0627 0.0624 0.0611 0.0629 0.0628 0.0625
D^max 0.1522 0.1521 0.1539 0.15311 0.1523 0.1528 0.1525 0.1524 0.1527 0.15310 0.1526
Total 211 448 5311 4710 355 469 242 242 386 407 242
n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
100 Bias( β^ ) 0.0033 -0.03911 -0.0139 -0.0126 0.0085 0.0127 0.0128 -0.02710 -0.0044 -0.0032 0.0021
RMSE( β^ ) 0.0873 0.1098 0.14810 0.15211 0.0986 0.0997 0.0861 0.0862 0.0965 0.1329 0.0884
Bias( μ^ ) 0.0264 0.03210 0.06711 0.0091 0.0309 0.0268 0.0266 0.0265 0.0267 0.0252 0.0253
RMSE( μ^ ) 0.2772 0.3099 0.34811 0.32710 0.2817 0.2796 0.2771 0.2773 0.2795 0.3028 0.2774
D^abs 0.0443 0.0446 0.0444 0.0448 0.04410 0.04411 0.0447 0.0431 0.0449 0.0442 0.0445
D^max 0.1111 0.1123 0.1127 0.1126 0.11310 0.11311 0.1125 0.1122 0.1129 0.1124 0.1128
Total 161 478 5211 427 478 5010 285 232 396 274 253
n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
200 Bias( β^ ) 0.0011 -0.03211 -0.0168 -0.02010 0.0033 0.0056 0.0057 -0.0179 -0.0034 -0.0035 0.0012
RMSE( β^ ) 0.0613 0.0838 0.11310 0.11311 0.0687 0.0676 0.0591 0.0602 0.0665 0.0929 0.0614
Bias( μ^ ) 0.0125 0.01810 0.05411 -0.0179 0.0148 0.0122 0.0124 0.0126 0.0123 0.0101 0.0127
RMSE( μ^ ) 0.1982 0.2249 0.26311 0.24110 0.1997 0.1986 0.1981 0.1984 0.1985 0.2148 0.1983
D^abs 0.0312 0.0314 0.03111 0.0313 0.0311 0.03110 0.0317 0.0318 0.0315 0.0319 0.0316
D^max 0.0812 0.0811 0.0828 0.0823 0.0826 0.08210 0.0825 0.08211 0.0824 0.0829 0.0827
Total 151 439 5911 4610 325 406 252 406 263 418 294
Overall Total 61 308 4311 3710 277 339 154 143 165 266 112

Table 3 Simulations results for μ = 0.2 and β = 2.0. 

n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
20 Bias( β^ ) 0.0273 -0.0444 0.0495 0.0607 0.0566 0.07010 0.0679 -0.07111 -0.0042 -0.0618 0.0011
RMSE( β^ ) 0.2162 0.2183 0.33310 0.34011 0.2568 0.2609 0.2215 0.2021 0.2367 0.2346 0.2204
Bias( μ^ ) 0.0117 0.0119 0.0022 0.02111 0.01310 0.0073 0.0094 0.0095 0.0106 0.0011 0.0118
RMSE( μ^ ) 0.1608 0.16810 0.1689 0.17411 0.1584 0.1583 0.1597 0.1571 0.1582 0.1596 0.1585
D^abs 0.0973 0.0976 0.0977 0.0975 0.0988 0.0972 0.0974 0.09811 0.0989 0.0971 0.09810
D^max 0.2225 0.2228 0.2226 0.2214 0.2212 0.2213 0.2227 0.22311 0.22210 0.2201 0.2229
Total 282 409 398 4911 387 303 364 409 364 231 376
n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
50 Bias( β^ ) 0.0085 -0.04511 -0.0031 0.0032 0.0196 0.0267 0.0288 -0.04110 -0.0033 -0.0399 0.0044
RMSE( β^ ) 0.1272 0.1457 0.21010 0.21011 0.1436 0.1468 0.1303 0.1261 0.1415 0.1519 0.1304
Bias( μ^ ) 0.0048 0.0059 0.00811 0.0022 0.00610 0.0011 0.0034 0.0035 0.0047 -0.0023 0.0046
RMSE( μ^ ) 0.0996 0.1079 0.11511 0.11310 0.1007 0.0995 0.0983 0.0982 0.0984 0.1008 0.0981
D^abs 0.0628 0.0626 0.0627 0.0625 0.0623 0.06310 0.0621 0.0629 0.06311 0.0622 0.0624
D^max 0.1536 0.1535 0.15310 0.1537 0.1523 0.1539 0.1521 0.1538 0.15411 0.1522 0.1534
Total 354 4710 5011 377 354 408 201 354 419 333 232
n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
100 Bias( β^ ) 0.0053 -0.04011 -0.0178 -0.0147 0.0094 0.0125 0.0146 -0.02610 -0.0021 -0.0249 0.0042
RMSE( β^ ) 0.0893 0.1099 0.15111 0.15110 0.0997 0.0976 0.0872 0.0861 0.0975 0.1078 0.0904
Bias( μ^ ) 0.0039 0.0027 0.01111 -0.00510 0.0016 0.0001 0.0014 0.0013 0.0012 -0.0015 0.0028
RMSE( μ^ ) 0.0692 0.0769 0.08511 0.08410 0.0707 0.0704 0.0705 0.0703 0.0706 0.0718 0.0691
D^abs 0.04411 0.0442 0.0445 0.04410 0.0441 0.0448 0.0449 0.0447 0.0443 0.0444 0.0446
D^max 0.11311 0.1123 0.1124 0.11310 0.1121 0.1138 0.1139 0.1136 0.1122 0.1135 0.1137
Total 397 419 5010 5711 262 325 356 304 191 397 283
n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
200 Bias( β^ ) 0.0022 -0.03311 -0.02010 -0.0179 0.0054 0.0076 0.0075 -0.0158 -0.0011 -0.0147 0.0033
RMSE( β^ ) 0.0623 0.0829 0.11311 0.11110 0.0697 0.0696 0.0601 0.0602 0.0685 0.0768 0.0624
Bias( μ^ ) 0.0005 0.0019 0.01111 -0.00710 0.0018 0.0004 -0.0003 0.0002 0.0001 -0.0016 0.0017
RMSE( μ^ ) 0.0492 0.0549 0.06411 0.06110 0.0507 0.0503 0.0505 0.0504 0.0506 0.0508 0.0491
D^abs 0.0313 0.0315 0.0317 0.0318 0.0316 0.0312 0.03210 0.0319 0.03211 0.0311 0.0314
D^abs 0.0826 0.0828 0.0825 0.0829 0.0823 0.0811 0.08210 0.0827 0.08311 0.0812 0.0824
Total 211 519 5510 5611 357 222 346 324 357 324 233
Overall Total 141 379 3910 4011 206 185 174 217 217 153 141

Table 4 Simulations results for μ = 0.8 and β = 0.5. 

n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
20 Bias( β^ ) 0.0284 -0.0465 0.07510 0.0728 0.0546 0.07911 0.0727 -0.0729 -0.0042 -0.0073 0.0031
RMSE( β^ ) 0.2132 0.2143 0.36911 0.36810 0.2567 0.2708 0.2285 0.1971 0.2406 0.3029 0.2284
Bias( μ^ ) -0.0292 -0.03610 -0.04911 -0.0316 -0.0261 -0.0303 -0.0315 -0.0359 -0.0328 -0.0304 -0.0327
RMSE( μ^ ) 0.1601 0.17810 0.18611 0.1748 0.1622 0.1623 0.1634 0.1667 0.1656 0.1749 0.1635
D^abs 0.0962 0.0978 0.0977 0.0973 0.09810 0.0976 0.0975 0.0961 0.09811 0.0989 0.0974
D^max 0.2202 0.2228 0.2215 0.2201 0.22310 0.2227 0.2216 0.2203 0.22311 0.2229 0.2204
Total 131 449 5511 365 365 387 324 303 449 438 252
n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
50 Bias( β^ ) 0.0116 -0.04711 0.0115 0.0042 0.0217 0.0309 0.0298 -0.04410 -0.0043 0.0011 0.0064
RMSE( β^ ) 0.1282 0.1467 0.21311 0.20810 0.1436 0.1488 0.1293 0.1221 0.1385 0.1889 0.1324
Bias( μ^ ) -0.0122 -0.0156 -0.01710 -0.02411 -0.0121 -0.0124 -0.0123 -0.0158 -0.0157 -0.0169 -0.0135
RMSE( μ^ ) 0.1001 0.1119 0.11410 0.12211 0.1025 0.1014 0.1003 0.1037 0.1036 0.1118 0.1002
D^abs 0.0623 0.0622 0.0621 0.0624 0.06210 0.0629 0.0625 0.0628 0.06311 0.0626 0.0627
D^max 0.1539 0.1521 0.1522 0.1537 0.1536 0.1534 0.1523 0.1535 0.15411 0.1538 0.15310
Total 231 365 397 4511 354 386 252 397 4310 419 323
n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE W LS
100 Bias( β^ ) 0.0054 -0.04111 -0.0127 -0.0116 0.0085 0.0149 0.0138 -0.02710 -0.0022 0.0011 0.0043
RMSE( β^ ) 0.0883 0.1108 0.15211 0.15010 0.0996 0.0997 0.0851 0.0862 0.0955 0.1329 0.0894
Bias( μ^ ) -0.0063 -0.00810 -0.0041 -0.01611 -0.0062 -0.0074 -0.0076 -0.0088 -0.0077 -0.0089 -0.0075
RMSE( μ^ ) 0.0725 0.0788 0.08410 0.08811 0.0713 0.0727 0.0726 0.0701 0.0712 0.0799 0.0714
D^abs 0.0447 0.0444 0.0449 0.0446 0.0441 0.04410 0.0445 0.0443 0.0442 0.04411 0.0448
D^max 0.1126 0.1124 0.1128 0.11310 0.1121 0.11311 0.1122 0.1125 0.1123 0.1139 0.1127
Total 283 457 468 5411 181 489 283 295 212 489 316
n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
200 Bias( β^ ) 0.0022 -0.03311 -0.01910 -0.0168 0.0045 0.0077 0.0066 -0.0179 -0.0011 0.0023 0.0034
RMSE( β^ ) 0.0613 0.0838 0.11411 0.11310 0.0686 0.0687 0.0591 0.0602 0.0685 0.0949 0.0624
Bias( μ^ ) -0.0044 -0.00510 0.0031 -0.01411 -0.0042 -0.0047 -0.0045 -0.0059 -0.0048 -0.0043 -0.0046
RMSE( μ^ ) 0.0491 0.0569 0.06110 0.06511 0.0504 0.0493 0.0492 0.0507 0.0505 0.0558 0.0506
D^abs 0.0312 0.0318 0.0319 0.0313 0.0315 0.0316 0.03110 0.0314 0.0317 0.03211 0.0311
D^max 0.0812 0.0827 0.0829 0.0824 0.0825 0.0828 0.08210 0.0823 0.0826 0.08211 0.0811
Total 141 5311 5010 479 273 387 345 345 324 458 222
Overall Total 61 328 3610 3610 132 297 144 205 256 349 132

Table 5 Simulations results for μ = 0.8 and β = 2.0. 

n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
20 Bias( β^ ) 0.0263 -0.0404 0.0556 0.0658 0.0495 0.07411 0.0699 -0.07110 -0.0082 -0.0647 0.0071
RMSE( β^ ) 0.2142 0.2163 0.33210 0.33711 0.2528 0.2639 0.2244 0.1971 0.2367 0.2346 0.2275
Bias( μ^ ) -0.0027 -0.0026 -0.00511 0.0002 -0.0013 -0.0039 -0.0028 -0.0024 -0.00310 -0.0001 -0.0025
RMSE( μ^ ) 0.0394 0.04210 0.04311 0.0419 0.0405 0.0392 0.0393 0.0391 0.0406 0.0408 0.0407
D^abs 0.0978 0.0974 0.0977 0.0962 0.09910 0.0973 0.0989 0.0976 0.0975 0.0961 0.09911
D^max 0.2217 0.2203 0.2218 0.2191 0.22310 0.2216 0.2229 0.2215 0.2214 0.2202 0.22411
Total 314 303 5311 335 419 407 4210 272 346 251 407
n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
50 Bias( β^ ) 0.0125 -0.04611 0.0032 0.0021 0.0206 0.0318 0.0257 -0.04410 -0.0064 -0.0399 0.0063
RMSE( β^ ) 0.1293 0.1497 0.20510 0.20511 0.1466 0.1509 0.1282 0.1251 0.1405 0.1498 0.1324
Bias( μ^ ) -0.0015 -0.0014 -0.0002 -0.00211 -0.0016 -0.0019 -0.0018 -0.0013 -0.0017 0.0001 -0.00110
RMSE( μ^ ) 0.0253 0.0279 0.02810 0.02811 0.0257 0.0242 0.0255 0.0241 0.0256 0.0258 0.0254
D^abs 0.0621 0.06310 0.0623 0.0624 0.06311 0.0628 0.0625 0.0627 0.0629 0.0626 0.0622
D^max 0.1521 0.15411 0.1524 0.1523 0.15310 0.1525 0.1538 0.1537 0.1539 0.1526 0.1522
Total 181 5211 314 418 4610 418 355 293 407 386 252
n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
100 Bias( β^ ) 0.0063 -0.03811 -0.0115 -0.0116 0.0114 0.0158 0.0127 -0.02610 -0.0032 -0.0249 0.0031
RMSE( β^ ) 0.0893 0.1119 0.15211 0.14910 0.1016 0.1027 0.0861 0.0862 0.0975 0.1088 0.0904
Bias( μ^ ) -0.0003 -0.0007 0.00110 -0.00311 -0.0001 -0.0004 -0.0006 -0.0002 -0.0005 0.0018 -0.0019
RMSE( μ^ ) 0.0171 0.0199 0.02110 0.02111 0.0176 0.0174 0.0175 0.0172 0.0177 0.0188 0.0173
D^abs 0.0441 0.04410 0.0448 0.0442 0.0449 0.0444 0.0443 0.0446 0.0447 0.0445 0.04511
D^max 0.1133 0.1138 0.1137 0.1121 0.11310 0.1122 0.1134 0.1136 0.1139 0.1135 0.11411
Total 141 5411 5110 418 366 294 262 283 355 439 397
n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
200 Bias( β^ ) 0.0022 -0.03111 -0.0158 -0.01510 0.0064 0.0076 0.0065 -0.0159 -0.0021 -0.0147 0.0033
RMSE( β^ ) 0.0633 0.0829 0.11411 0.11210 0.0707 0.0706 0.0601 0.0602 0.0685 0.0778 0.0634
Bias( μ^ ) -0.0003 -0.0004 0.00210 -0.00311 -0.0002 -0.0008 -0.0006 -0.0001 -0.0005 0.0009 -0.0007
RMSE( μ^ ) 0.0121 0.0149 0.01510 0.01611 0.0127 0.0124 0.0123 0.0122 0.0125 0.0126 0.0128
D^abs 0.0319 0.0314 0.03111 0.0316 0.0318 0.0317 0.0311 0.0312 0.03110 0.0315 0.0313
D^max 0.08210 0.0814 0.08211 0.0826 0.0829 0.0812 0.0813 0.0828 0.0815 0.0811 0.0827
Total 284 419 6111 5410 367 273 201 325 262 346 378
Overall Total 101 3410 3611 318 329 225 183 132 204 225 247

Table 6 Overall performance of estimation methods. 

Scenario AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
(μ = 0.2, β = 0.5) 61 308 4311 3710 277 339 154 143 165 266 112
(μ = 0.2, β = 1.5) 193 3910 4011 318 193 318 193 207 132 193 91
(μ = 0.2, β = 2.0) 141 379 3910 4011 206 185 174 217 217 153 141
(μ = 0.4, β = 0.5) 91 329 4010 4111 236 267 123 102 185 288 174
(μ = 0.4, β = 1.5) 122 329 4111 3910 298 277 133 205 216 133 111
(μ = 0.4, β = 2.0) 236 287 4211 329 329 195 174 298 61 112 153
(μ = 0.6, β = 0.5) 185 319 3510 3911 237 174 206 163 142 288 131
(μ = 0.6, β = 1.5) 257 369 3810 3810 112 348 236 123 101 134 205
(μ = 0.6, β = 2.0) 81 3610 359 4011 256 267 153 164 225 267 122
(μ = 0.8, β = 0.5) 61 328 3610 3610 132 297 144 205 256 349 132
(μ = 0.8, β = 1.5) 91 3610 339 3911 153 195 142 277 184 298 195
(μ = 0.8, β = 2.0) 101 3410 3611 318 329 225 183 132 204 225 247
Total 1591 4039 45811 44310 2697 3018 1973 2185 2044 2646 1782

Table 7 Simulations results of interval estimation for μ = 0.2 and β = 0.5. 

n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
20 CP( β^ ) 0.9445 0.9098 0.9463 0.9444 0.9297 0.9099 0.90310 0.85511 0.9512 0.9586 0.9511
CP( μ^ ) 0.9492 0.9628 0.93210 0.92311 0.9397 0.9406 0.9415 0.9629 0.9523 0.9594 0.9511
AW( β^ ) 0.4213 0.3982 0.72710 0.72911 0.5107 0.5438 0.4445 0.3351 0.4506 0.5899 0.4334
AW( μ^ ) 0.4643 0.52111 0.4826 0.5179 0.4684 0.4512 0.4451 0.5058 0.4837 0.52010 0.4765
Total 132 297 297 3511 255 255 214 297 183 297 111
n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
50 CP( β^ ) 0.9492 0.86411 0.9494 0.9467 0.9466 0.9388 0.9349 0.87810 0.9493 0.9475 0.9501
CP( μ^ ) 0.9457 0.96211 0.9502 0.9465 0.94310 0.9448 0.9466 0.9569 0.9483 0.9501 0.9474
AW( β^ ) 0.2493 0.2595 0.40711 0.40610 0.2857 0.2908 0.2482 0.2171 0.2716 0.3689 0.2554
AW( μ^ ) 0.3063 0.35410 0.37011 0.3489 0.3095 0.3032 0.3001 0.3217 0.3126 0.3368 0.3074
Total 152 3711 288 3110 288 266 183 277 183 235 131
n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
100 CP( β^ ) 0.9513 0.84711 0.9358 0.9309 0.9455 0.9456 0.9407 0.89610 0.9502 0.9524 0.9501
CP( μ^ ) 0.9483 0.96311 0.96010 0.9569 0.9492 0.9475 0.9475 0.9537 0.9511 0.9548 0.9484
AW( β^ ) 0.1733 0.1946 0.29210 0.29211 0.1957 0.1958 0.1692 0.1571 0.1895 0.2599 0.1754
AW( μ^ ) 0.2173 0.2529 0.28611 0.25310 0.2205 0.2172 0.2151 0.2247 0.2216 0.2388 0.2184
Total 121 379 3910 3910 195 216 154 257 143 298 132
n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
200 CP( β^ ) 0.9577 0.82911 0.9278 0.91010 0.9544 0.9532 0.9501 0.9139 0.9556 0.9555 0.9532
CP( μ^ ) 0.9459 0.9524 0.9501 0.9485 0.9485 0.94510 0.9468 0.9483 0.9477 0.9492 0.94411
AW( β^ ) 0.1213 0.1458 0.21511 0.21510 0.1357 0.1356 0.1172 0.1121 0.1335 0.1839 0.1224
AW( μ^ ) 0.1542 0.1789 0.21511 0.18610 0.1555 0.1544 0.1531 0.1567 0.1556 0.1688 0.1543
Total 214 3210 319 3511 214 226 121 202 247 247 202
Overall Total 92 3710 349 4211 225 236 123 236 164 278 61

Table 8 Simulations results of interval estimation for μ = 0.2 and β = 2.0. 

n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
20 CP( β^ ) 0.9395 0.9058 0.9584 0.9491 0.9346 0.9157 0.9059 0.85711 0.9472 0.90110 0.9553
CP( μ^ ) 0.9405 0.9616 0.9359 0.92111 0.93310 0.9404 0.9358 0.9603 0.9481 0.9627 0.9472
AW( β^ ) 1.6714 1.5752 2.49310 2.50611 1.9688 2.0419 1.7436 1.3371 1.7737 1.6273 1.7055
AW( μ^ ) 0.1244 0.14411 0.1388 0.1399 0.1233 0.1212 0.1191 0.1377 0.1306 0.14310 0.1285
Total 183 277 3110 3211 277 224 246 224 162 309 151
n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
50 CP( β^ ) 0.9462 0.86511 0.9406 0.9455 0.9463 0.9387 0.9238 0.87610 0.9463 0.9099 0.9491
CP( μ^ ) 0.9457 0.96511 0.9546 0.9482 0.9458 0.9459 0.9483 0.95610 0.9535 0.9534 0.9511
AW( β^ ) 0.9993 1.0375 1.61210 1.62411 1.1448 1.1629 0.9972 0.8701 1.0857 1.0826 1.0224
AW( μ^ ) 0.0773 0.0909 0.09411 0.09110 0.0784 0.0772 0.0761 0.0817 0.0796 0.0848 0.0785
Total 153 3611 3310 288 235 276 142 288 214 276 111
n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
100 CP( β^ ) 0.9474 0.84411 0.9307 0.9269 0.9474 0.9501 0.9376 0.89710 0.9501 0.9278 0.9493
CP( μ^ ) 0.9458 0.96511 0.9559 0.9536 0.9491 0.9484 0.94410 0.9523 0.9525 0.9547 0.9482
AW( β^ ) 0.6933 0.7756 1.16210 1.16711 0.7817 0.7828 0.6782 0.6271 0.7565 0.7929 0.7034
AW( μ^ ) 0.0543 0.0639 0.07011 0.06710 0.0555 0.0542 0.0541 0.0567 0.0556 0.0588 0.0544
Total 185 3710 3710 369 173 152 196 217 173 328 131
n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
200 CP( β^ ) 0.9545 0.82911 0.9278 0.91510 0.9484 0.9483 0.9456 0.9189 0.9501 0.9367 0.9492
CP( μ^ ) 0.9447 0.96110 0.9474 0.9501 0.9448 0.9429 0.93811 0.9502 0.9466 0.9513 0.9475
AW( β^ ) 0.4853 0.5809 0.85910 0.86111 0.5427 0.5416 0.4712 0.4501 0.5315 0.5758 0.4904
AW( μ^ ) 0.0383 0.0449 0.05111 0.04910 0.0395 0.0384 0.0381 0.0397 0.0396 0.0408 0.0382
Total 182 3911 3310 329 247 226 205 194 182 268 131
Overall Total 133 3910 4011 379 226 184 195 237 112 318 41

Table 9 Simulations results of interval estimation for μ = 0.8 and β = 0.5. 

n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
20 CP( β^ ) 0.9445 0.9078 0.9426 0.9464 0.9357 0.9069 0.90110 0.85911 0.9533 0.9521 0.9482
CP( μ^ ) 0.9462 0.9585 0.92510 0.92111 0.9279 0.9396 0.9368 0.9627 0.9444 0.9443 0.9491
AW( β^ ) 0.4233 0.3952 0.73511 0.73110 0.5117 0.5448 0.4445 0.3341 0.4526 0.5949 0.4344
AW( μ^ ) 0.4654 0.52711 0.5209 0.4907 0.4553 0.4522 0.4471 0.5118 0.4856 0.52510 0.4795
Total 142 267 3611 3210 267 256 245 279 193 234 121
n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
50 CP( β^ ) 0.9463 0.86411 0.9446 0.9417 0.9464 0.9308 0.9239 0.87710 0.9532 0.9511 0.9464
CP( μ^ ) 0.9457 0.95810 0.9535 0.9492 0.94111 0.9459 0.9456 0.9558 0.9474 0.9473 0.9501
AW( β^ ) 0.2503 0.2585 0.41111 0.40810 0.2867 0.2918 0.2492 0.2161 0.2716 0.3709 0.2564
AW( μ^ ) 0.3054 0.35410 0.3499 0.37111 0.3033 0.3022 0.3001 0.3237 0.3136 0.3388 0.3075
Total 172 3611 3110 309 256 278 183 267 183 215 141
n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
100 CP( β^ ) 0.9492 0.84111 0.9319 0.9368 0.9511 0.9455 0.9427 0.89510 0.9534 0.9576 0.9483
CP( μ^ ) 0.9439 0.96111 0.9533 0.9544 0.9446 0.94110 0.9448 0.9566 0.9491 0.9455 0.9492
AW( β^ ) 0.1733 0.1936 0.29210 0.29211 0.1957 0.1968 0.1692 0.1571 0.1895 0.2609 0.1764
AW( μ^ ) 0.2173 0.2539 0.25410 0.28511 0.2175 0.2172 0.2151 0.2257 0.2216 0.2388 0.2174
Total 173 3711 329 3410 195 257 184 246 162 288 131
n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
200 CP( β^ ) 0.9536 0.82811 0.9258 0.91410 0.9475 0.9474 0.9492 0.9149 0.9547 0.9483 0.9501
CP( μ^ ) 0.9482 0.95610 0.9469 0.95811 0.9485 0.9491 0.9468 0.9524 0.9537 0.9483 0.9476
AW( β^ ) 0.1213 0.1458 0.21510 0.21511 0.1357 0.1356 0.1172 0.1121 0.1335 0.1839 0.1224
AW( μ^ ) 0.1543 0.1789 0.18610 0.21511 0.1555 0.1544 0.1531 0.1577 0.1556 0.1678 0.1542
Total 143 3810 379 4311 226 154 131 215 258 237 131
Overall Total 102 399 399 4011 245 257 133 278 164 245 41

Table 10 Simulations results of interval estimation for μ = 0.8 and β = 2.0. 

n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
20 CP( β^ ) 0.9445 0.9078 0.9533 0.9532 0.9356 0.9087 0.9039 0.86711 0.9511 0.90110 0.9464
CP( μ^ ) 0.9444 0.9637 0.93110 0.91911 0.9329 0.9425 0.9406 0.9648 0.9501 0.9553 0.9462
AW( β^ ) 1.6704 1.5842 2.50510 2.51311 1.9588 2.0479 1.7466 1.3391 1.7677 1.6223 1.7115
AW( μ^ ) 0.1244 0.14411 0.1399 0.1367 0.1233 0.1212 0.1191 0.1368 0.1306 0.14310 0.1275
Total 173 288 3211 3110 266 235 224 288 151 266 162
n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
50 CP( β^ ) 0.9434 0.85811 0.9462 0.9425 0.9396 0.9287 0.9278 0.87310 0.9471 0.9129 0.9443
CP( μ^ ) 0.9455 0.96011 0.9511 0.9492 0.9439 0.9457 0.94110 0.9578 0.9464 0.9556 0.9483
AW( β^ ) 1.0023 1.0365 1.62511 1.62310 1.1458 1.1679 0.9952 0.8661 1.0827 1.0826 1.0234
AW( μ^ ) 0.0773 0.0909 0.09010 0.09311 0.0774 0.0762 0.0761 0.0817 0.0796 0.0848 0.0785
Total 151 3611 245 289 278 256 214 267 183 2910 151
n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
100 CP( β^ ) 0.9464 0.85111 0.9328 0.9337 0.9463 0.9336 0.9425 0.89510 0.9511 0.9259 0.9482
CP( μ^ ) 0.9513 0.9559 0.95610 0.9535 0.9484 0.9492 0.9467 0.95911 0.9458 0.9536 0.9501
AW( β^ ) 0.6943 0.7776 1.17011 1.16910 0.7827 0.7848 0.6772 0.6261 0.7555 0.7939 0.7024
AW( μ^ ) 0.0543 0.0639 0.06610 0.06911 0.0554 0.0542 0.0541 0.0567 0.0556 0.0588 0.0555
Total 132 3510 3911 339 184 184 153 297 206 328 121
n Qtd AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
200 CP( β^ ) 0.9464 0.84111 0.9308 0.9169 0.9482 0.9445 0.9446 0.91410 0.9491 0.9307 0.9473
CP( μ^ ) 0.9477 0.95911 0.9511 0.9524 0.9485 0.9548 0.9483 0.9526 0.9512 0.95610 0.9469
AW( β^ ) 0.4853 0.5819 0.86310 0.86311 0.5437 0.5406 0.4692 0.4501 0.5315 0.5758 0.4894
AW( μ^ ) 0.0382 0.0449 0.04910 0.05111 0.0395 0.0384 0.0381 0.0397 0.0396 0.0408 0.0383
Total 163 4011 298 3510 194 236 121 247 142 339 194
Overall Total 92 4011 359 3810 226 215 123 297 123 338 81

Table 11 Overall performance of estimation methods with respect the interval estimation. 

Scenario AD AD2 AD2L AD2R ADR CvM MLE MPS OLS PCE WLS
(μ = 0.2 = 0.5) 92 3710 349 4211 225 236 123 236 164 278 61
(μ = 0.2 = 1.5) 61 3910 4211 349 267 298 123 245 154 256 72
(μ = 0.2 = 2.0) 133 3910 4011 379 226 184 195 237 112 318 41
(μ = 0.4 = 0.5) 92 3910 389 3910 235 277 103 277 164 246 71
(μ = 0.4 = 1.5) 81 4211 369 4110 246 257 143 268 195 143 92
(μ = 0.4 = 2.0) 113 4010 4010 379 298 205 71 267 124 216 82
(μ = 0.6 = 0.5) 92 3810 339 4211 266 307 144 307 113 205 51
(μ = 0.6 = 1.5) 196 4111 379 4010 143 196 143 288 155 122 91
(μ = 0.6 = 2.0) 51 3810 369 4111 277 205 173 308 194 216 62
(μ = 0.8 = 0.5) 102 399 399 4011 245 257 133 278 164 245 41
(μ = 0.8 = 1.5) 51 359 359 4111 256 245 112 298 123 277 144
(μ = 0.8 = 2.0) 92 4011 359 3810 226 215 123 297 123 338 81
Total 1132 46710 4459 47211 2847 2816 1553 3228 1744 2795 871

In Tables 2-5 we report the empirical values of (20)-(23). A superscript indicates the rank of each of the estimators among all the estimators for that metric. For example, Table 2 presents he bias of the MLE ( β^ ) in the first row as 0.0709 for n = 20. This indicates that the bias of β^ obtained using the method of maximum likelihood ranks 9 th among all other estimators.

The following observations can be drawn from Tables 2-5.

  • 1. All the estimators reveal the property of consistency, i.e., the RMSE decreases when the sample size increases.

  • 2. The bias of β^ decreases when n increases for all estimation methods.

  • 3. The bias of μ^ decreases when n increases for all estimation methods.

  • 4. The bias of μ^ generally decreases when β increases for any given β and n for all estimation methods.

  • 5. The bias of β^ generally decreases when μ increases for any given μ and n for all estimation methods.

  • 6. D^abs is smaller than D^max for all estimation techniques. Again, these statistics become smaller when n increases.

The overall ranks of the estimation methods are presented in Table 6. For the parameter combinations considered in our study, Anderson-Darling estimator (AD) turns out to the best (overall score of 159) in the overall ranking closely followed by the method of weighted least square (WLS) (overall score of 178).

In the previous tables, we have obtained the point estimates of each method of estimation. However, it is also important to know the behaviour of interval estimation for each method of estimations. Therefore, we computed the parametric Bootstrap confidence interval 12 and evaluate their coverage probability and average length of the simulated confidence intervals. The results are presented in Tables 7-10.

From the results reported in Tables 7-10, it is observed that as sample sizes increases, the coverage probability increases for both the parameters as well as for the estimation methods, while the average width of the confidence intervals decreases as the sample sizes increases for both the parameters and estimation methods.

The overall positions of the interval estimates are presented in Table 11. It is observed that WLS is the best method for interval estimation based on parametric Boostrap confidence intervals. The next best method is the AD, followed by MLE.

Thus, based on our study we may conclude that AD and WLS are the best methods for estimating the parameters of unit-Logistic distribution for both point and interval estimation. Therefore, we suggest to use AD and WLS methods of estimation for practical purposes.

4 ILLUSTRATIVE EXAMPLES

In this section, the performance of the eleven estimation methods is compared through two real data applications.

The first data (data set I) is available in software R and corresponds to 48 observations of twelve core samples from petroleum reservoirs that were sampled by four cross-sections. The second data set (data set-II) can be found in 4 and represents the total milk production in the first birth of 107 cows from SINDI race.

The parameter estimates and their corresponding Bootstrap confidence intervals for all estimation methods considered are summarized in Tables 12 and 13. We also present the results of formal goodness-of-fit tests, the Kolmogorov-Smirnov (KS) test, in order to show that the unit-Logistic distribution can be used to model these two data sets.

Table 12 Parameter estimates, 95% confidence intervals based on parametric Bootstrap and K-S test: data set I. 

Method μ LCL UCL β LCL UCL KS (p-value)
MLE 0.2033 0.1847 0.2240 3.8276 3.0530 5.0733 0.0979 (0.7469)
MPS 0.2019 0.1792 0.2236 3.5707 2.6905 4.3026 0.0907 (0.8242)
PCE 0.2058 0.1808 0.2298 3.2550 2.3271 4.0154 0.1114 (0.5907)
OLS 0.2014 0.1792 0.2226 3.6391 2.7666 4.7659 0.0879 (0.8520)
WLS 0.2034 0.1814 0.2249 3.7032 2.8428 4.8165 0.0992 (0.7326)
CvM 0.2013 0.1793 0.2228 3.7596 3.0188 5.0458 0.0865 (0.8649)
AD 0.2034 0.1834 0.2257 3.7135 2.9679 4.7314 0.0990 (0.7351)
ADR 0.2018 0.1798 0.2262 3.4077 2.6352 4.6341 0.0991 (0.7335)
AD2R 0.2002 0.1736 0.2273 3.1894 2.0618 4.8186 0.1192 (0.5030)
AD2L 0.1958 0.1788 0.2163 4.9271 3.2415 7.3709 0.1379 (0.3205)
AD2 0.2103 0.1864 0.2363 3.6283 2.4818 4.5025 0.1374 (0.3249)

L(U)CL lower (upper) confidence limit.

Table 13 Parameter estimates, 95% confidence intervals based on parametric Bootstrap and KS test: data set II. 

Method μ LCL UCL β LCL UCL KS (p-value)
MLE 0.4729 0.4317 0.5148 1.9103 1.6565 2.2716 0.0571 (0.8767)
MPS 0.4723 0.4019 0.5355 1.8338 1.3818 2.2098 0.0618 (0.8081)
PCE 0.4686 0.4037 0.5298 1.9449 1.4745 2.3979 0.0580 (0.8642)
OLS 0.4789 0.4166 0.5341 2.0873 1.5868 2.7337 0.0695 (0.6788)
WLS 0.4762 0.4137 0.5331 2.0752 1.5931 2.6993 0.0682 (0.7026)
CvM 0.2013 0.1793 0.2228 3.7596 3.0188 5.0458 0.7350 (0.0000)
AD 0.4733 0.4304 0.5163 1.9681 1.6871 2.3073 0.0610 (0.8201)
ADR 0.4798 0.4441 0.5181 2.2317 1.8949 2.7488 0.0765 (0.5584)
AD2R 0.4876 0.4442 0.5254 2.4237 1.7670 3.1201 0.0848 (0.4254)
AD2L 0.4981 0.4313 0.5785 1.3952 0.9988 1.7848 0.1480 (0.0184)
AD2 0.4419 0.3867 0.4974 1.6351 1.2378 1.8874 0.1207 (0.0886)

L(U)CL lower (upper) confidence limit.

From Table 12 we can see that all estimates provide a good fit to the data set. It is also observed that the AD2L and MPS estimators give the shortest confidence intervals for μ and β, respectively.

The results in Table 13 indicate that the CvM estimates do not provide a good fit to this data set as per KS statistic is concerned. It is also observed that MLE has the lowest value of KS. It is also noteworthy, that MLE and ADR have the shortest confidence intervals for μ and β.

5 CONCLUDING REMARKS

In this paper, we have performed an extensive simulation study to compare eleven aforementioned methods of estimation. We have compared estimators with respect to bias, root meansquared error, the average absolute difference between the theoretical and empirical estimate of the distribution functions, and the maximum absolute difference between the theoretical and empirical distribution functions. We have also calculated the coverage probability and the average width of the Bootstrap confidence intervals. We have also compared estimators by two real data applications. The simulation results show that AD estimators is the best performing estimator in terms of biases and RMSE. The next best performing estimators is the WLS estimators, followed by MLE. The real data applications show that the AD2L and MPS estimators give the shortest confidence intervals for μ and β, respectively for the data set I and MLE and ADR have the shortest confidence intervals for the data set II. Hence, we can argue that the AD estimators, weighted least squares estimators, AD2L, MPS, ADR and ML estimators are among the best performing estimators for unit-logistic distribution.

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Received: November 16, 2017; Accepted: October 12, 2018

*Corresponding author.

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