Bayes Estimation of Topp-Leone Distribution Under Symmetric Entropy Loss Function Based on Lower Record Values
Department of Basic Subjects, Hunan University of Finance and Economics, Changsha, China
To cite this article:
Lanping Li. Bayes Estimation of Topp-Leone Distribution Under Symmetric Entropy Loss Function Based on Lower Record Values. Science Journal of Applied Mathematics and Statistics. Vol. 4, No. 6, 2016, pp. 284-288. doi: 10.11648/j.sjams.20160406.16
Received: October 9, 2016; Accepted: October 20, 2016; Published: November 14, 2016
Abstract:This paper will study the estimation of parameter of Topp-Leone distribution based on lower record values. First, the minimum variance unbiased estimator and maximum likelihood estimator are obtained. Then the Bayes estimator is derived under symmetric loss function and further the empirical Bayes estimators is also obtained based on marginal probability density of record sample and maximum likelihood method. Finally, the admissibility and inadmissibility of a generally class of inverse linear estimators are also discussed.
Keywords: Admissibility, Bayes and Empirical Bayes Estimators, Record Values, Symmetric Entropy Loss Function
Topp-Leone distribution was introduced by Topp and Leone in 1955 and used it to model some failure data. Ghitany and Kotz  pointed out Topp-Leone distribution is useful for modelling life-time phenomena. They studied some reliability measures of this distribution such as the hazard rate, mean residual life, reversed hazard rate, expected inactivity time, and their stochastic orderings. Al-Zahrani proposed a class of goodness-of-fit tests for the Topp-Leone distribution with estimated parameters and the proposed were based on the empirical distribution function, Kolomogorov-Smirnov, Cramer-von-Mises, etc. Sindhua  estimated the parameter of Topp-Leone distribution based on trimmed samples using Bayesian approach. Al-Zahrani and Alshomrani  studied the estimation of reliability of a stress-strength model in which the stress and strength samples are both distributed with Topp-Leone distribution. Bayoud  studied the estimation of the shape parameter of the Topp-Leone distribution from a view of classical and Bayesian approaches based on Type I censored samples. Feroze et al.  considered the Bayesian point and interval estimations of the two-component mixture of Topp-Leone distribution based on type I censored samples under a class of priors using four loss functions. El-Sayed et al.  used Bayesian and non-Bayesian approach to estimate coefficient of variation of Topp-Leone distribution under an adaptive Type-II Progressive Censoring samples.
This paper is devoted to study the admissibility and inadmissibility of the estimators of the unknown parameter of the Topp-Leone distribution based on lower record values. The cumulative distribution function and probability density function of Topp-Leone distribution are given as follows, respectively ():
Here is the unknown parameter.
Record values model, proposed by Chandler , is very related to the order statistics model, both of which are widely used in many real life applications involving data relating to weather, sports, economics, and life tests. sport, weather and life testing studies. Many authors have studied the statistical inference of various distributions based on record values and associated statistics. For example, Raqab  derived Exact expressions for single and product moments of three parameter generalized exponential distribution based record values. Soliman et al.  studied the Bayesian estimation in the context of record statistics values from the two-parameter Weibull distribution with respect to both symmetric loss function (squared error loss), and asymmetric loss function (LINEX) loss function. Jaheen  derived the Bayes estimators for the two unknown parameters of the Gompertz model using Laplace approximation under the squared error and LINEX loss functions. Ahmadi et al.  studied the Bayesian estimation and the admissibility of some estimators for the two parameters of the exponential distribution based on k-record values under LINEX loss function. More references can be found in [14-19].
The study of the admissible properties of estimators is an important topic in statistical inference. Yu and Sun  study the Bayes estimation and admissibility of the obtained estimators of Normal Distribution under q-Symmetric Entropy Loss Function. Bayoud  derived the Bayes and empirical Bayes estimates of the unknown parameter of Topp–Leone distribution under non informative and suitable conjugate priors. They proved that the proposed estimates are minimax and admissible. Zakerzadeh and Zahraie  studied the admissibility and inadmissibility of estimators of non-regular family under squared-log error loss function. Cao and kong  studied the admissibility of a class of linear estimators in a general multivariate linear model under balanced loss function.
The purpose of this paper is to study the estimation and admissible characters of estimators of the parameter of Topp-Leone distribution. The organization of this paper is as follows. Section 2 derives the maximum likelihood (ML) estimator and minimum variance unbiased estimator of the parameter of Topp-Leone distribution. Section 3 discusses the Bayes and empirical Bayes estimators of the parameter . Section 4 studies the admissibility and inadmissibility of a class of inverse linear estimators. Finally, Section 5 gives the conclusion of this paper.
2. ML and Minimum Variance Unbiased Estimation
This section first recalls some definitions of lower record values, then derives the ML estimator and minimum variance unbiased estimator of the parameter of Topp-Leone distribution.
Definition 1.  Let be a sequence of independent and identically distributed random variables. Suppose
We say , is a lower record value of this sequence.
Suppose be n lower record values from Topp-Leone distribution (1). is the observation of . When given the observation of. the likelihood function can be obtained as follows (Arnold et al. )
Then we have
The MLE can be obtained by solving the log-likelihood equation
Then the MLE can be derived as
The probability density function of is given by
Then we can get
According to Eq. (5), the probability density function of can be easily derived as
It means that distributed with Gamma distribution .
Further more, we have
thus is an unbiased estimator of . According to Eq. (3), is a sufficient statistics for the first n lower record values. The estimator is the minimum variance unbiased estimator of by using Lehmann and Scheffe’s theorem. That is
3. Bayes Estimation
Prior distribution and loss function are very important parts in Bayesian analysis. Though squared error loss is the most often used loss function in Bayesian inference, but in recent years, more loss functions, such as LINEX loss, squared log error loss function, entropy loss function and symmetric entropy loss function are put forward and applied in Bayesian estimation. This section will derive the Bayes estimatior under the following symmetric entropy loss function (Zhao et al. )
The Bayes estimator under the symmetric entropy loss is denoted by , given by
Assume that the parameter has the Gamma prior distribution with prior parameters and , noted by, . Then the prior probability density function of parameter is
Remark 1. The limiting case of the prior Gamma distribution is non-informative prior .
According to Eq. (3) and Eq. (10), using Bayesian theorem, we can get the posterior probability density function as follows:
Then is a posterior probability density of, i.e. .
The Bayes estimator of under symmetric entropy loss function is
which belongs to the form
Remark 2. When the parameter has the non-informative prior , the posterior distribution of is . Then we obtain the generalized Bayes estimator
Remark 3. When the prior parameter is unknown, we need estimate it using the empirical Bayes approach. From Eq. (3) and Eq. (10), the marginal probability density function of can be derived as
Based on , The ML estimator of is .
Substitute for in the Bayes estimator, then the empirical Bayes estimator is obtained as
Obviously, the empirical Bayes estimator also has the inverse linear form .
4. Admissibility of (cT+d)-1
The estimators obtained in Section 2 and Section 3 belong to the inverse linear estimators with the form . This section will discuss the admissibility and inadmissibility of estimators. In the following discussion, we always supposes that:
(i) are n (n>1) lower record values from Topp-Leone distribution (1).
(ii) Let be the observation of , and be the observation of .
In the rest of this section, all obtained estimators will be compared on the basis of their risks under the entropy loss function. We will also study the conditions under which general inverse linear estimators are admissible in terms of risk function.
Theorem 1. If and , then the estimator is admissible.
Proof. From Eq. (12), it always true that the coefficient of is strictly between 0 and for any and , and the constant is strictly larger than 0. Then the estimator is admissible for and .
If , then it is easy to verify that the estimator is also admissible since it is the unique Bayes estimator for which when .
Theorem 2. Let be the parameter space and be the action space. The inverse linear estimator is inadmissible under the symmetric entropy loss function (8) whenever one of the following conditions holds:
(i) or ;
Proof. (i) When or , then we note that takes on negative values with positive probability. Then the estimator is dominated by the estimator. Thus is inadmissible under case (i).
(ii) Under symmetric loss function, the risk function of the estimator is
Then derivative of the risk with respect to is
, where ,
Thus the risk of the estimator is minimized at . Then the estimator is dominated by the estimator , and thus is inadmissible under case (ii).
(iii). Under the symmetric loss function, the difference of risk functions between the inverse linear estimator and are as follows:
In the condition (iii), , then
Therefore, is minimized at . That is to say, is dominated by in this case. Then the estimator is inadmissible under the case (iii).
Remark 4. By Theorem 2, the ML estimator and the empirical Bayes estimator are both inadmissible. They are both dominated by the Bayes estimator .
This paper derived the ML estimator, minimum variance unbiased estimator, Bayes estimator and empirical Bayes estimator of the parameter of the Topp-Leone distribution on based on lower record values. These estimators all belong to inverse linear estimators with the form . The admissibility and inadmissibility of are studied under different conditions. As a result, ML estimator and the minimum variance unbiased estimator are admissible.
This study is partially supported by Natural Science Foundation of Hunan Province (No. 2015JJ3030) and Foundation of Hunan Educational Committee (No.15C0228). The author also gratefully acknowledges the helpful comments and suggestions of the reviewers, which have improved the presentation.