THE UNIT-LOGISTIC DISTRIBUTION: DIFFERENT METHODS OF ESTIMATION

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:

$X=g^{-1}\left(\frac{Y-\gamma }{\delta }\right),$ (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:

$g\left(X\right)=\log \left(\frac{X}{1-X}\right),$ (2)

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

$f(x|\gamma ,\delta )=\frac{\delta e^{\gamma }{x}^{\delta -1}(1-x{)}^{\delta -1}}{{\left[x^{\delta }{e}^{\gamma }+(1-x{)}^{\delta }\right]}^2},0<x<1.$ (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:

$f(x|\mu ,\beta )=\frac{\beta {\mu }^{\beta }{x}^{\beta -1}{(1-\mu )}^{\beta }(1-x{)}^{\beta -1}}{{\left[(1-\mu {)}^{\beta }{x}^{\beta }+{\mu }^{\beta }(1-x{)}^{\beta }\right]}^2},0<x<1$ (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:

$F(x|\mu ,\beta )={\left[1+{\left(\frac{\mu (1-x)}{x(1-\mu )}\right)}^{\beta }\right]}^{-1},0<x<1$ (5)

and

$Q(p|\mu ,\beta )=\frac{\mu p^{1/\beta }}{(1-\mu )(1-p{)}^{1/\beta }+\mu p^{1/\beta }},0<p<1.$ (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:

$$\begin{gathered} \mathcal{l}=(\mu ,\beta |x)=n\log \beta +n\beta \log (1-\mu )+n\beta \log \mu \\ +(\beta -1)\sum _{i=1}^n\log x_i+(\beta -1)\sum _{i=1}^n\log (1-x_i) \\ -2\sum _{i=1}^n\log \left[(1-\mu {)}^{\beta }{x_i}^{\beta }+{\mu }^{\beta }(1-x_i{)}^{\beta }\right]. \end{gathered}$$ (7)

The maximum likelihood estimates ${\hat{\mu }}_{MLE}$ and ${\hat{\beta }}_{MLE}$ of the parameters μ and β, respectively, can be obtained by maximizing (7), or equivalently solving the following normal equations:

$\frac{\partial \mathcal{l}}{\partial \beta }=\frac{n\left(\beta -\mu \beta \right)}{\mu \left(1-\mu \right)}-2\beta \sum _{i=1}^n\frac{{(1-\mu )}^{\beta -1}{x}_i^{\beta }-{\mu }^{\beta -1}{\left(1-x_i\right)}^{\beta }}{{(1-\mu )}^{\beta }{x}_i^{\beta }+{\mu }^{\beta }{\left(1-x_i\right)}^{\beta }}=0,$ (8)

$$\begin{gathered} \frac{\partial \mathcal{l}}{\partial \beta }=\frac{n}{\beta }+n\log (1-\mu )+n\log \mu +\sum _{i=1}^n\log x_i+\sum _{i=1}^n\log (1-x_i) \\ -2\sum _{i=1}^n\frac{{x_i}^{\beta }\left[\log (1-\mu )+\log x_i\right](1-\mu )\beta +2{\mu }^{\beta }(1-x_i{)}^{\beta }\left[\log (1-x_i)+\log \mu \right]}{(1-\mu {)}^{\beta }{x_i}^{\beta }+{\mu }^{\beta }(1-x_i{)}^{\beta }}=0. \end{gathered}$$ (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:

$$\begin{gathered} D_i(\mu ,\beta )=F(x_{\left(i\right)}|\mu ,\beta )-F(x_{(i-1)}|\mu ,\beta ),i=1,...,n, \\ F(x_{\left(0\right)}|\mu ,\beta )=0\text{and}F(x_{(n+1)}|\mu ,\beta )=0. \end{gathered}$$

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

From ^{2), (3}, the MPSEs, ${\hat{\mu }}_{MPS}$ and ${\hat{\beta }}_{MPS}$ are the values of μ and β, which maximize the geometric mean of the spacings:

$G(\mu ,\beta |x)={\left[\prod _{i=1}^{n+1}{D}_i\left(\mu ,\beta \right)\right]}^{\frac{1}{n+1}}$ (10)

or, equivalently, by maximize its logarithm:

$H(\mu ,\beta |x)=\frac{1}{n+1}\sum _{i=1}^{n+1}\log D_i.$ (11)

The estimators ${\hat{\mu }}_{MPS}$ and ${\hat{\beta }}_{MPS}$ of the parameters α and β can also be obtained by solving the nonlinear equations:

$$\begin{gathered} \frac{\partial }{\partial \mu }H(\mu ,\beta )=\frac{1}{n+1}\sum _{i=1}^{n+1}\frac{1}{D_i(\mu ,\beta )}\left[{\Delta }_1(x_{\left(i\right)}|\mu ,\beta )-{\Delta }_1(x_{(i-1)}|\mu ,\beta )\right]=0, \\ \frac{\partial }{\partial \beta }H(\mu ,\beta )=\frac{1}{n+1}\sum _{i=1}^{n+1}\frac{1}{D_i(\mu ,\beta )}\left[{\Delta }_2(x_{\left(i\right)}|\mu ,\beta )-{\Delta }_2(x_{(i-1)}|\mu ,\beta )\right]=0, \end{gathered}$$

where

${\Delta }_1(x_{\left(i\right)}|\mu ,\beta )=\frac{{\mu }^{\beta -1}\beta x_{\left(i\right)}^{-\beta }{\left({\displaystyle \frac{\mu -1}{x_{\left(i\right)}-1}}\right)}^{\beta }}{(\mu -1){\left[x_{\left(i\right)}{\mu }^{-\beta }{\left(\frac{\mu -1}{x_{\left(i\right)}-1}\right)}^{\beta }+1\right]}^2}$ (12)

and

${\Delta }_2(x_{i:n}|\mu ,\beta )=\frac{\left[\log \mu +\log (1-x_{\left(i\right)})-\log x_{\left(i\right)}-\log (1-\mu )\right]x_{\left(i\right)}^{-\beta }{\mu }^{\beta }{\left({\displaystyle \frac{\mu -1}{x_{\left(i\right)}-1}}\right)}^{\beta }}{(\mu -1){\left[x_{\left(i\right)}{\mu }^{-\beta }{\left(\frac{\mu -1}{x_{\left(i\right)}-1}\right)}^{\beta }+1\right]}^2}.$ (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, ${\hat{\mu }}_{PCE}$ and ${\hat{\beta }}_{PCE}$ can be obtained by minimizing, with respect to μ and β, the nonlinear function:

$P(\mu ,\beta |x)=\sum _{i=1}^n{\left[x_{\left(i\right)}-\frac{\mu p_i^{1/\beta }}{}(1-\mu )(1-p_i{)}^{1/\beta }+\mu p_i^{1/\beta }\right]}^2,$ (14)

where $p_i=\frac{i}{n+1}$ 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, n − i + 1). Therefore, we have:

$E\left[F\left(X_{\left(i\right)}\right)\right]=\frac{i}{n+1}\text{and}V\left[F\left(X_{\left(i\right)}\right)\right]=\frac{i\left(n-i+1\right)}{{\left(n+1\right)}^2\left(n+2\right)}$ (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 ${\hat{\mu }}_{OLS}$ and ${\hat{\beta }}_{OLS}$ of the parameters μ and β can be obtained by minimizing the function:

$S(\mu ,\beta |x)=\sum _{i=1}^n{\left[F\left(x_{\left(i\right)}|\mu ,\beta \right)-\frac{i}{n+1}\right]}^2$ (16)

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

$$\begin{gathered} \sum _{i=1}^n\left[F\left(x_{\left(i\right)}|\mu ,\beta \right)-\frac{i}{n+1}\right]{\Delta }_1\left(x_{\left(i\right)}|\mu ,\beta \right)=0, \\ \sum _{i=1}^n\left[F\left(x_{\left(i\right)}|\mu ,\beta \right)-\frac{{\displaystyle i}}{{\displaystyle n+1}}\right]{\Delta }_2\left(x_{\left(i\right)}|\mu ,\beta \right)=0 \end{gathered}$$

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 ${\hat{\mu }}_{WLS}$ and ${\hat{\beta }}_{WLS}$, respectively are obtained by minimizing the function:

$W(\mu ,\beta |x)=\sum _{i=1}^n\frac{(n+1{)}^2(n+2)}{i(n-i+1)}{\left[F\left(x_{\left(i\right)}|\mu ,\beta \right)-\frac{i}{n+1}\right]}^2$ (17)

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

$$\begin{gathered} \sum _{i=1}^n\frac{{\displaystyle {\left(n+1\right)}^2}{\displaystyle }{\displaystyle \left(n+2\right)}}{i\left(n-i+1\right)}\left[F\left(x_{\left(i\right)}|\mu ,\beta \right)-\frac{i}{n+1}\right]{\Delta }_1\left(x_{\left(i\right)}|\mu ,\beta \right)=0, \\ \sum _{i=1}^n\frac{{\displaystyle {\left(n+1\right)}^2\left(n+2\right)}}{i\left(n-i+1\right)}\left[F\left(x_{\left(i\right)}|\mu ,\beta \right)-\frac{i}{n+1}\right]{\Delta }_2\left(x_{\left(i\right)}|\mu ,\beta \right)=0 \end{gathered}$$

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	$w_n^2=\frac{1}{12n}+\sum _{i=1}^n{\left(x_{\left(i\right)}-\frac{2i-1}{2n}\right)}^2$
AD	$A_n^2=-n-\frac{1}{n}+\sum _{i=1}^n\left(2i-1\right)\left[\log x_{\left(i\right)}+\log \left(1-x_{\left(n+1-i\right)}\right)\right]$
ADR	$R_n^2=\frac{n}{2}-2\sum _{i=1}^n{x}_{\left(i\right)}-\frac{1}{n}\sum _{i=1}^n\left(2i-1\right)\log \left(1-x_{\left(n+1-i\right)}\right)$
AD2R	$r_n^2=2\sum _{i=1}^n\log \left(1-x_{\left(i\right)}\right)+\frac{1}{n}\sum _{i=1}^n\frac{2i-1}{1-x_{\left(n+1-i\right)}}$
AD2L	$l_n^2=2\sum _{i=1}^n\log x_{\left(i\right)}+\frac{1}{n}\sum _{i=1}^n\frac{2i-1}{x_{\left(i\right)}}$
AD2	$a_n^2=2\sum _{i=1}^n\left[\log x_{\left(i\right)}+\log (1-x_{\left(i\right)})\right]+\frac{1}{n}\sum _{i=1}^n\left(\frac{2i-1}{x_{\left(i\right)}}+\frac{2i-1}{1-x_{\left(n+1-i\right)}}\right)$

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 ${\hat{\mu }}_{CvM}$ and ${\hat{\beta }}_{CvM}$ are obtained by minimizing with respect to μ and β the function:

$C(\mu ,\beta |x)=\frac{1}{12n}+\sum _{i=1}^n{\left(F\left(x_{\left(i\right)}|\mu ,\beta \right)-\frac{2i-1}{2n}\right)}^2.$ (18)

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

$$\begin{gathered} \sum _{i=1}^n\left(F\left(x_{\left(i\right)}|\mu ,\beta \right)-\frac{{\displaystyle 2i-1}}{{\displaystyle 2n}}\right){\Delta }_1\left(x_{\left(i\right)}|\mu ,\beta \right)=0, \\ \sum _{i=1}^n\left(F\left(x_{\left(i\right)}|\mu ,\beta \right)-\frac{{\displaystyle 2i-1}}{{\displaystyle 2n}}\right){\Delta }_2\left(x_{\left(i\right)}|\mu ,\beta \right)=0 \end{gathered}$$

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, ${\hat{\mu }}_{ADE}$ and ${\hat{\beta }}_{ADE}$ by minimizing the function

$A(\mu ,\beta |x)=-n-\frac{1}{n}\sum _{i=1}^n(2i-1)\left\{\log F\left(x_{\left(i\right)}|\mu ,\beta \right)+\log \overline{F}\left(x_{(n+1-i)}|\mu ,\beta \right)\right\}.$ (19)

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

$$\begin{gathered} \sum _{i=1}^n\left(2i-1\right)\left[\frac{{\Delta }_1\left(x_{\left(i\right)}|\mu ,\beta \right)}{F\left(x_{\left(i\right)}|\mu ,\beta \right)}-\frac{{\Delta }_1\left(x_{\left(n+1-i\right)}|\mu ,\beta \right)}{\overline{F}\left(x_{\left(n+1-i\right)}|\mu ,\beta \right)}\right]=0, \\ \sum _{i=1}^n\left(2i-1\right)\left[\frac{{\Delta }_2\left(x_{\left(i\right)}|\mu ,\beta \right)}{F\left(x_{\left(i\right)}|\mu ,\beta \right)}-\frac{{\Delta }_2\left(x_{\left(n+1-i\right)}|\mu ,\beta \right)}{\overline{F}\left(x_{\left(n+1-i\right)}|\mu ,\beta \right)}\right]=0 \end{gathered}$$

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:

$\mathrm{Bias}\left(\hat{\Theta }\right)=\frac{1}{M}\sum _{i=1}^M\left(\frac{{\hat{\Theta }}_i-\Theta }{\Theta }\right),$ (20)

$\mathrm{RMSE}\left(\hat{\Theta }\right)=\sqrt{\frac{1}{M}\sum _{i=1}^M{\left({\hat{\Theta }}_i-\Theta \right)}^2},$ (21)

$D_{abs}=\frac{1}{M\times n}\sum _{i=1}^M\sum _{j=1}^n|F\left(y_{ij}|\Theta \right)-F\left(y_{ij}|\hat{\Theta }\right)|,$ (22)

$D_{max}=\frac{1}{M}\sum _{i=1}^M\underset{j}{\text{max}}|F\left(y_{ij}|\Theta \right)-F\left(y_{ij}|\hat{\Theta }\right)|.$ (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($\hat{\beta }$)	0.0254	-0.0385	0.07110	0.0697	0.0526	0.07711	0.0709	-0.0708	-0.0093	0.0001	0.0022
	RMSE($\hat{\beta }$)	0.2142	0.2163	0.36110	0.36611	0.2587	0.2718	0.2265	0.2001	0.2356	0.2989	0.2234
	Bias($\hat{\mu }$)	0.1083	0.1269	0.0881	0.16711	0.12910	0.1097	0.1072	0.1084	0.1096	0.1118	0.1085
	RMSE($\hat{\mu }$)	0.6422	0.69310	0.6768	0.72211	0.6577	0.6456	0.6411	0.6423	0.6444	0.6819	0.6445
	${\hat{D}}_{abs}$	0.0988	0.0972	0.09811	0.0973	0.09810	0.0989	0.0987	0.0975	0.0961	0.0976	0.0974
	${\hat{D}}_{max}$	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($\hat{\beta }$)	0.0076	-0.04611	0.0012	-0.0011	0.0167	0.0269	0.0248	-0.04410	-0.0075	-0.0044	0.0023
	RMSE($\hat{\beta }$)	0.1243	0.1437	0.20110	0.20511	0.1426	0.1458	0.1242	0.1221	0.1375	0.1879	0.1274
	Bias($\hat{\mu }$)	0.0466	0.05910	0.07611	0.0463	0.0559	0.0478	0.0464	0.0465	0.0477	0.0441	0.0462
	RMSE($\hat{\mu }$)	0.3982	0.4439	0.46011	0.45510	0.4057	0.4006	0.3981	0.3983	0.4005	0.4328	0.3994
	${\hat{D}}_{abs}$	0.0622	0.0626	0.06210	0.06211	0.0623	0.0627	0.0624	0.0611	0.0629	0.0628	0.0625
	${\hat{D}}_{max}$	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($\hat{\beta }$)	0.0033	-0.03911	-0.0139	-0.0126	0.0085	0.0127	0.0128	-0.02710	-0.0044	-0.0032	0.0021
	RMSE($\hat{\beta }$)	0.0873	0.1098	0.14810	0.15211	0.0986	0.0997	0.0861	0.0862	0.0965	0.1329	0.0884
	Bias($\hat{\mu }$)	0.0264	0.03210	0.06711	0.0091	0.0309	0.0268	0.0266	0.0265	0.0267	0.0252	0.0253
	RMSE($\hat{\mu }$)	0.2772	0.3099	0.34811	0.32710	0.2817	0.2796	0.2771	0.2773	0.2795	0.3028	0.2774
	${\hat{D}}_{abs}$	0.0443	0.0446	0.0444	0.0448	0.04410	0.04411	0.0447	0.0431	0.0449	0.0442	0.0445
	${\hat{D}}_{max}$	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($\hat{\beta }$)	0.0011	-0.03211	-0.0168	-0.02010	0.0033	0.0056	0.0057	-0.0179	-0.0034	-0.0035	0.0012
	RMSE($\hat{\beta }$)	0.0613	0.0838	0.11310	0.11311	0.0687	0.0676	0.0591	0.0602	0.0665	0.0929	0.0614
	Bias($\hat{\mu }$)	0.0125	0.01810	0.05411	-0.0179	0.0148	0.0122	0.0124	0.0126	0.0123	0.0101	0.0127
	RMSE($\hat{\mu }$)	0.1982	0.2249	0.26311	0.24110	0.1997	0.1986	0.1981	0.1984	0.1985	0.2148	0.1983
	${\hat{D}}_{abs}$	0.0312	0.0314	0.03111	0.0313	0.0311	0.03110	0.0317	0.0318	0.0315	0.0319	0.0316
	${\hat{D}}_{max}$	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($\hat{\beta }$)	0.0273	-0.0444	0.0495	0.0607	0.0566	0.07010	0.0679	-0.07111	-0.0042	-0.0618	0.0011
	RMSE($\hat{\beta }$)	0.2162	0.2183	0.33310	0.34011	0.2568	0.2609	0.2215	0.2021	0.2367	0.2346	0.2204
	Bias($\hat{\mu }$)	0.0117	0.0119	0.0022	0.02111	0.01310	0.0073	0.0094	0.0095	0.0106	0.0011	0.0118
	RMSE($\hat{\mu }$)	0.1608	0.16810	0.1689	0.17411	0.1584	0.1583	0.1597	0.1571	0.1582	0.1596	0.1585
	${\hat{D}}_{abs}$	0.0973	0.0976	0.0977	0.0975	0.0988	0.0972	0.0974	0.09811	0.0989	0.0971	0.09810
	${\hat{D}}_{max}$	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($\hat{\beta }$)	0.0085	-0.04511	-0.0031	0.0032	0.0196	0.0267	0.0288	-0.04110	-0.0033	-0.0399	0.0044
	RMSE($\hat{\beta }$)	0.1272	0.1457	0.21010	0.21011	0.1436	0.1468	0.1303	0.1261	0.1415	0.1519	0.1304
	Bias($\hat{\mu }$)	0.0048	0.0059	0.00811	0.0022	0.00610	0.0011	0.0034	0.0035	0.0047	-0.0023	0.0046
	RMSE($\hat{\mu }$)	0.0996	0.1079	0.11511	0.11310	0.1007	0.0995	0.0983	0.0982	0.0984	0.1008	0.0981
	${\hat{D}}_{abs}$	0.0628	0.0626	0.0627	0.0625	0.0623	0.06310	0.0621	0.0629	0.06311	0.0622	0.0624
	${\hat{D}}_{max}$	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($\hat{\beta }$)	0.0053	-0.04011	-0.0178	-0.0147	0.0094	0.0125	0.0146	-0.02610	-0.0021	-0.0249	0.0042
	RMSE($\hat{\beta }$)	0.0893	0.1099	0.15111	0.15110	0.0997	0.0976	0.0872	0.0861	0.0975	0.1078	0.0904
	Bias($\hat{\mu }$)	0.0039	0.0027	0.01111	-0.00510	0.0016	0.0001	0.0014	0.0013	0.0012	-0.0015	0.0028
	RMSE($\hat{\mu }$)	0.0692	0.0769	0.08511	0.08410	0.0707	0.0704	0.0705	0.0703	0.0706	0.0718	0.0691
	${\hat{D}}_{abs}$	0.04411	0.0442	0.0445	0.04410	0.0441	0.0448	0.0449	0.0447	0.0443	0.0444	0.0446
	${\hat{D}}_{max}$	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($\hat{\beta }$)	0.0022	-0.03311	-0.02010	-0.0179	0.0054	0.0076	0.0075	-0.0158	-0.0011	-0.0147	0.0033
	RMSE($\hat{\beta }$)	0.0623	0.0829	0.11311	0.11110	0.0697	0.0696	0.0601	0.0602	0.0685	0.0768	0.0624
	Bias($\hat{\mu }$)	0.0005	0.0019	0.01111	-0.00710	0.0018	0.0004	-0.0003	0.0002	0.0001	-0.0016	0.0017
	RMSE($\hat{\mu }$)	0.0492	0.0549	0.06411	0.06110	0.0507	0.0503	0.0505	0.0504	0.0506	0.0508	0.0491
	${\hat{D}}_{abs}$	0.0313	0.0315	0.0317	0.0318	0.0316	0.0312	0.03210	0.0319	0.03211	0.0311	0.0314
	${\hat{D}}_{abs}$	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($\hat{\beta }$)	0.0284	-0.0465	0.07510	0.0728	0.0546	0.07911	0.0727	-0.0729	-0.0042	-0.0073	0.0031
	RMSE($\hat{\beta }$)	0.2132	0.2143	0.36911	0.36810	0.2567	0.2708	0.2285	0.1971	0.2406	0.3029	0.2284
	Bias($\hat{\mu }$)	-0.0292	-0.03610	-0.04911	-0.0316	-0.0261	-0.0303	-0.0315	-0.0359	-0.0328	-0.0304	-0.0327
	RMSE($\hat{\mu }$)	0.1601	0.17810	0.18611	0.1748	0.1622	0.1623	0.1634	0.1667	0.1656	0.1749	0.1635
	${\hat{D}}_{abs}$	0.0962	0.0978	0.0977	0.0973	0.09810	0.0976	0.0975	0.0961	0.09811	0.0989	0.0974
	${\hat{D}}_{max}$	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($\hat{\beta }$)	0.0116	-0.04711	0.0115	0.0042	0.0217	0.0309	0.0298	-0.04410	-0.0043	0.0011	0.0064
	RMSE($\hat{\beta }$)	0.1282	0.1467	0.21311	0.20810	0.1436	0.1488	0.1293	0.1221	0.1385	0.1889	0.1324
	Bias($\hat{\mu }$)	-0.0122	-0.0156	-0.01710	-0.02411	-0.0121	-0.0124	-0.0123	-0.0158	-0.0157	-0.0169	-0.0135
	RMSE($\hat{\mu }$)	0.1001	0.1119	0.11410	0.12211	0.1025	0.1014	0.1003	0.1037	0.1036	0.1118	0.1002
	${\hat{D}}_{abs}$	0.0623	0.0622	0.0621	0.0624	0.06210	0.0629	0.0625	0.0628	0.06311	0.0626	0.0627
	${\hat{D}}_{max}$	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($\hat{\beta }$)	0.0054	-0.04111	-0.0127	-0.0116	0.0085	0.0149	0.0138	-0.02710	-0.0022	0.0011	0.0043
	RMSE($\hat{\beta }$)	0.0883	0.1108	0.15211	0.15010	0.0996	0.0997	0.0851	0.0862	0.0955	0.1329	0.0894
	Bias($\hat{\mu }$)	-0.0063	-0.00810	-0.0041	-0.01611	-0.0062	-0.0074	-0.0076	-0.0088	-0.0077	-0.0089	-0.0075
	RMSE($\hat{\mu }$)	0.0725	0.0788	0.08410	0.08811	0.0713	0.0727	0.0726	0.0701	0.0712	0.0799	0.0714
	${\hat{D}}_{abs}$	0.0447	0.0444	0.0449	0.0446	0.0441	0.04410	0.0445	0.0443	0.0442	0.04411	0.0448
	${\hat{D}}_{max}$	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($\hat{\beta }$)	0.0022	-0.03311	-0.01910	-0.0168	0.0045	0.0077	0.0066	-0.0179	-0.0011	0.0023	0.0034
	RMSE($\hat{\beta }$)	0.0613	0.0838	0.11411	0.11310	0.0686	0.0687	0.0591	0.0602	0.0685	0.0949	0.0624
	Bias($\hat{\mu }$)	-0.0044	-0.00510	0.0031	-0.01411	-0.0042	-0.0047	-0.0045	-0.0059	-0.0048	-0.0043	-0.0046
	RMSE($\hat{\mu }$)	0.0491	0.0569	0.06110	0.06511	0.0504	0.0493	0.0492	0.0507	0.0505	0.0558	0.0506
	${\hat{D}}_{abs}$	0.0312	0.0318	0.0319	0.0313	0.0315	0.0316	0.03110	0.0314	0.0317	0.03211	0.0311
	${\hat{D}}_{max}$	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($\hat{\beta }$)	0.0263	-0.0404	0.0556	0.0658	0.0495	0.07411	0.0699	-0.07110	-0.0082	-0.0647	0.0071
	RMSE($\hat{\beta }$)	0.2142	0.2163	0.33210	0.33711	0.2528	0.2639	0.2244	0.1971	0.2367	0.2346	0.2275
	Bias($\hat{\mu }$)	-0.0027	-0.0026	-0.00511	0.0002	-0.0013	-0.0039	-0.0028	-0.0024	-0.00310	-0.0001	-0.0025
	RMSE($\hat{\mu }$)	0.0394	0.04210	0.04311	0.0419	0.0405	0.0392	0.0393	0.0391	0.0406	0.0408	0.0407
	${\hat{D}}_{abs}$	0.0978	0.0974	0.0977	0.0962	0.09910	0.0973	0.0989	0.0976	0.0975	0.0961	0.09911
	${\hat{D}}_{max}$	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($\hat{\beta }$)	0.0125	-0.04611	0.0032	0.0021	0.0206	0.0318	0.0257	-0.04410	-0.0064	-0.0399	0.0063
	RMSE($\hat{\beta }$)	0.1293	0.1497	0.20510	0.20511	0.1466	0.1509	0.1282	0.1251	0.1405	0.1498	0.1324
	Bias($\hat{\mu }$)	-0.0015	-0.0014	-0.0002	-0.00211	-0.0016	-0.0019	-0.0018	-0.0013	-0.0017	0.0001	-0.00110
	RMSE($\hat{\mu }$)	0.0253	0.0279	0.02810	0.02811	0.0257	0.0242	0.0255	0.0241	0.0256	0.0258	0.0254
	${\hat{D}}_{abs}$	0.0621	0.06310	0.0623	0.0624	0.06311	0.0628	0.0625	0.0627	0.0629	0.0626	0.0622
	${\hat{D}}_{max}$	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($\hat{\beta }$)	0.0063	-0.03811	-0.0115	-0.0116	0.0114	0.0158	0.0127	-0.02610	-0.0032	-0.0249	0.0031
	RMSE($\hat{\beta }$)	0.0893	0.1119	0.15211	0.14910	0.1016	0.1027	0.0861	0.0862	0.0975	0.1088	0.0904
	Bias($\hat{\mu }$)	-0.0003	-0.0007	0.00110	-0.00311	-0.0001	-0.0004	-0.0006	-0.0002	-0.0005	0.0018	-0.0019
	RMSE($\hat{\mu }$)	0.0171	0.0199	0.02110	0.02111	0.0176	0.0174	0.0175	0.0172	0.0177	0.0188	0.0173
	${\hat{D}}_{abs}$	0.0441	0.04410	0.0448	0.0442	0.0449	0.0444	0.0443	0.0446	0.0447	0.0445	0.04511
	${\hat{D}}_{max}$	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($\hat{\beta }$)	0.0022	-0.03111	-0.0158	-0.01510	0.0064	0.0076	0.0065	-0.0159	-0.0021	-0.0147	0.0033
	RMSE($\hat{\beta }$)	0.0633	0.0829	0.11411	0.11210	0.0707	0.0706	0.0601	0.0602	0.0685	0.0778	0.0634
	Bias($\hat{\mu }$)	-0.0003	-0.0004	0.00210	-0.00311	-0.0002	-0.0008	-0.0006	-0.0001	-0.0005	0.0009	-0.0007
	RMSE($\hat{\mu }$)	0.0121	0.0149	0.01510	0.01611	0.0127	0.0124	0.0123	0.0122	0.0125	0.0126	0.0128
	${\hat{D}}_{abs}$	0.0319	0.0314	0.03111	0.0316	0.0318	0.0317	0.0311	0.0312	0.03110	0.0315	0.0313
	${\hat{D}}_{max}$	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($\hat{\beta }$)	0.9445	0.9098	0.9463	0.9444	0.9297	0.9099	0.90310	0.85511	0.9512	0.9586	0.9511
	CP($\hat{\mu }$)	0.9492	0.9628	0.93210	0.92311	0.9397	0.9406	0.9415	0.9629	0.9523	0.9594	0.9511
	AW($\hat{\beta }$)	0.4213	0.3982	0.72710	0.72911	0.5107	0.5438	0.4445	0.3351	0.4506	0.5899	0.4334
	AW($\hat{\mu }$)	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($\hat{\beta }$)	0.9492	0.86411	0.9494	0.9467	0.9466	0.9388	0.9349	0.87810	0.9493	0.9475	0.9501
	CP($\hat{\mu }$)	0.9457	0.96211	0.9502	0.9465	0.94310	0.9448	0.9466	0.9569	0.9483	0.9501	0.9474
	AW($\hat{\beta }$)	0.2493	0.2595	0.40711	0.40610	0.2857	0.2908	0.2482	0.2171	0.2716	0.3689	0.2554
	AW($\hat{\mu }$)	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($\hat{\beta }$)	0.9513	0.84711	0.9358	0.9309	0.9455	0.9456	0.9407	0.89610	0.9502	0.9524	0.9501
	CP($\hat{\mu }$)	0.9483	0.96311	0.96010	0.9569	0.9492	0.9475	0.9475	0.9537	0.9511	0.9548	0.9484
	AW($\hat{\beta }$)	0.1733	0.1946	0.29210	0.29211	0.1957	0.1958	0.1692	0.1571	0.1895	0.2599	0.1754
	AW($\hat{\mu }$)	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($\hat{\beta }$)	0.9577	0.82911	0.9278	0.91010	0.9544	0.9532	0.9501	0.9139	0.9556	0.9555	0.9532
	CP($\hat{\mu }$)	0.9459	0.9524	0.9501	0.9485	0.9485	0.94510	0.9468	0.9483	0.9477	0.9492	0.94411
	AW($\hat{\beta }$)	0.1213	0.1458	0.21511	0.21510	0.1357	0.1356	0.1172	0.1121	0.1335	0.1839	0.1224
	AW($\hat{\mu }$)	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($\hat{\beta }$)	0.9395	0.9058	0.9584	0.9491	0.9346	0.9157	0.9059	0.85711	0.9472	0.90110	0.9553
	CP($\hat{\mu }$)	0.9405	0.9616	0.9359	0.92111	0.93310	0.9404	0.9358	0.9603	0.9481	0.9627	0.9472
	AW($\hat{\beta }$)	1.6714	1.5752	2.49310	2.50611	1.9688	2.0419	1.7436	1.3371	1.7737	1.6273	1.7055
	AW($\hat{\mu }$)	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($\hat{\beta }$)	0.9462	0.86511	0.9406	0.9455	0.9463	0.9387	0.9238	0.87610	0.9463	0.9099	0.9491
	CP($\hat{\mu }$)	0.9457	0.96511	0.9546	0.9482	0.9458	0.9459	0.9483	0.95610	0.9535	0.9534	0.9511
	AW($\hat{\beta }$)	0.9993	1.0375	1.61210	1.62411	1.1448	1.1629	0.9972	0.8701	1.0857	1.0826	1.0224
	AW($\hat{\mu }$)	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($\hat{\beta }$)	0.9474	0.84411	0.9307	0.9269	0.9474	0.9501	0.9376	0.89710	0.9501	0.9278	0.9493
	CP($\hat{\mu }$)	0.9458	0.96511	0.9559	0.9536	0.9491	0.9484	0.94410	0.9523	0.9525	0.9547	0.9482
	AW($\hat{\beta }$)	0.6933	0.7756	1.16210	1.16711	0.7817	0.7828	0.6782	0.6271	0.7565	0.7929	0.7034
	AW($\hat{\mu }$)	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($\hat{\beta }$)	0.9545	0.82911	0.9278	0.91510	0.9484	0.9483	0.9456	0.9189	0.9501	0.9367	0.9492
	CP($\hat{\mu }$)	0.9447	0.96110	0.9474	0.9501	0.9448	0.9429	0.93811	0.9502	0.9466	0.9513	0.9475
	AW($\hat{\beta }$)	0.4853	0.5809	0.85910	0.86111	0.5427	0.5416	0.4712	0.4501	0.5315	0.5758	0.4904
	AW($\hat{\mu }$)	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($\hat{\beta }$)	0.9445	0.9078	0.9426	0.9464	0.9357	0.9069	0.90110	0.85911	0.9533	0.9521	0.9482
	CP($\hat{\mu }$)	0.9462	0.9585	0.92510	0.92111	0.9279	0.9396	0.9368	0.9627	0.9444	0.9443	0.9491
	AW($\hat{\beta }$)	0.4233	0.3952	0.73511	0.73110	0.5117	0.5448	0.4445	0.3341	0.4526	0.5949	0.4344
	AW($\hat{\mu }$)	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($\hat{\beta }$)	0.9463	0.86411	0.9446	0.9417	0.9464	0.9308	0.9239	0.87710	0.9532	0.9511	0.9464
	CP($\hat{\mu }$)	0.9457	0.95810	0.9535	0.9492	0.94111	0.9459	0.9456	0.9558	0.9474	0.9473	0.9501
	AW($\hat{\beta }$)	0.2503	0.2585	0.41111	0.40810	0.2867	0.2918	0.2492	0.2161	0.2716	0.3709	0.2564
	AW($\hat{\mu }$)	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($\hat{\beta }$)	0.9492	0.84111	0.9319	0.9368	0.9511	0.9455	0.9427	0.89510	0.9534	0.9576	0.9483
	CP($\hat{\mu }$)	0.9439	0.96111	0.9533	0.9544	0.9446	0.94110	0.9448	0.9566	0.9491	0.9455	0.9492
	AW($\hat{\beta }$)	0.1733	0.1936	0.29210	0.29211	0.1957	0.1968	0.1692	0.1571	0.1895	0.2609	0.1764
	AW($\hat{\mu }$)	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($\hat{\beta }$)	0.9536	0.82811	0.9258	0.91410	0.9475	0.9474	0.9492	0.9149	0.9547	0.9483	0.9501
	CP($\hat{\mu }$)	0.9482	0.95610	0.9469	0.95811	0.9485	0.9491	0.9468	0.9524	0.9537	0.9483	0.9476
	AW($\hat{\beta }$)	0.1213	0.1458	0.21510	0.21511	0.1357	0.1356	0.1172	0.1121	0.1335	0.1839	0.1224
	AW($\hat{\mu }$)	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($\hat{\beta }$)	0.9445	0.9078	0.9533	0.9532	0.9356	0.9087	0.9039	0.86711	0.9511	0.90110	0.9464
	CP($\hat{\mu }$)	0.9444	0.9637	0.93110	0.91911	0.9329	0.9425	0.9406	0.9648	0.9501	0.9553	0.9462
	AW($\hat{\beta }$)	1.6704	1.5842	2.50510	2.51311	1.9588	2.0479	1.7466	1.3391	1.7677	1.6223	1.7115
	AW($\hat{\mu }$)	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($\hat{\beta }$)	0.9434	0.85811	0.9462	0.9425	0.9396	0.9287	0.9278	0.87310	0.9471	0.9129	0.9443
	CP($\hat{\mu }$)	0.9455	0.96011	0.9511	0.9492	0.9439	0.9457	0.94110	0.9578	0.9464	0.9556	0.9483
	AW($\hat{\beta }$)	1.0023	1.0365	1.62511	1.62310	1.1458	1.1679	0.9952	0.8661	1.0827	1.0826	1.0234
	AW($\hat{\mu }$)	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($\hat{\beta }$)	0.9464	0.85111	0.9328	0.9337	0.9463	0.9336	0.9425	0.89510	0.9511	0.9259	0.9482
	CP($\hat{\mu }$)	0.9513	0.9559	0.95610	0.9535	0.9484	0.9492	0.9467	0.95911	0.9458	0.9536	0.9501
	AW($\hat{\beta }$)	0.6943	0.7776	1.17011	1.16910	0.7827	0.7848	0.6772	0.6261	0.7555	0.7939	0.7024
	AW($\hat{\mu }$)	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($\hat{\beta }$)	0.9464	0.84111	0.9308	0.9169	0.9482	0.9445	0.9446	0.91410	0.9491	0.9307	0.9473
	CP($\hat{\mu }$)	0.9477	0.95911	0.9511	0.9524	0.9485	0.9548	0.9483	0.9526	0.9512	0.95610	0.9469
	AW($\hat{\beta }$)	0.4853	0.5819	0.86310	0.86311	0.5437	0.5406	0.4692	0.4501	0.5315	0.5758	0.4894
	AW($\hat{\mu }$)	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,β = 0.5)	92	3710	349	4211	225	236	123	236	164	278	61
(μ = 0.2,β = 1.5)	61	3910	4211	349	267	298	123	245	154	256	72
(μ = 0.2,β = 2.0)	133	3910	4011	379	226	184	195	237	112	318	41
(μ = 0.4,β = 0.5)	92	3910	389	3910	235	277	103	277	164	246	71
(μ = 0.4,β = 1.5)	81	4211	369	4110	246	257	143	268	195	143	92
(μ = 0.4,β = 2.0)	113	4010	4010	379	298	205	71	267	124	216	82
(μ = 0.6,β = 0.5)	92	3810	339	4211	266	307	144	307	113	205	51
(μ = 0.6,β = 1.5)	196	4111	379	4010	143	196	143	288	155	122	91
(μ = 0.6,β = 2.0)	51	3810	369	4111	277	205	173	308	194	216	62
(μ = 0.8,β = 0.5)	102	399	399	4011	245	257	133	278	164	245	41
(μ = 0.8,β = 1.5)	51	359	359	4111	256	245	112	298	123	277	144
(μ = 0.8,β = 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 ($\hat{\beta }$) in the first row as 0.070⁹ for n = 20. This indicates that the bias of $\hat{\beta }$ 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 $\hat{\beta }$ decreases when n increases for all estimation methods.

• 3. The bias of $\hat{\mu }$ decreases when n increases for all estimation methods.

• 4. The bias of $\hat{\mu }$ generally decreases when β increases for any given β and n for all estimation methods.

• 5. The bias of $\hat{\beta }$ generally decreases when μ increases for any given μ and n for all estimation methods.

• 6. ${\hat{D}}_{abs}$ is smaller than ${\hat{D}}_{max}$ 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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Publication Dates

• Publication in this collection
  Sep-Dec 2018

History

• Received
  16 Nov 2017
• Accepted
  12 Oct 2018