New biased estimators under the LINEX loss function

In this paper, using the asymmetric LINEX loss function we derive the risk function of the generalized Liu estimator and almost unbiased generalized Liu estimator. We also examine the risk performance of the feasible generalized Liu estimator and feasible almost unbiased generalized Liu estimator when the LINEX loss function is used.     

Comparison of risks of estimators under the LINEX loss for a family of truncated distributions

In most cases, we use a symmetric loss such as the quadratic loss in a usual estimation problem. But, in the non-regular case when the regularity conditions do not necessarily hold, it seems to be more reasonable to choose an asymmetric loss than the symmetric one. In this paper, we consider the Bayes estimation under the linear exponential (LINEX) loss which is regarded as a typical example of asymmetric loss. We also compare the Bayes risks of estimators under the LINEX loss for a family of truncated distributions and a location parameter family of truncated distributions.     

On the admissibility of ordered poisson parameter under the entropy loss function

Let X ₁, X ₂be two independent Poisson random variables with means θ ₁and θ ₂respectively. Assume 0 ≤ θ ₁ ≤ θ ₂ ≤ ∞. The problem is estimation of the ordered parameters which has received considerable attention in statistical literature during the last two decades, In this paper the main portion of the study is devoted to the problem of estimating the smallest of the two ordered Poisson means, when it is known which population corresponds to each mean under the entropy loss function. An extension of the problem to the k-ordered Poisson means is also discussed.     

E-Bayesian estimation of the exponentiated distribution family parameter under LINEX loss function

This paper is concerned with using the E-Bayesian method for computing estimates of the exponentiated distribution family parameter. Based on the LINEX loss function, formulas of E-Bayesian estimation for unknown parameter are given, these estimates are derived based on a conjugate prior. Moreover, property of E-Bayesian estimation—the relationship between of E-Bayesian estimations under different prior distributions of the hyper parameters are also provided. A comparison between the new method and the corresponding maximum likelihood techniques is conducted using the Monte Carlo simulation. Finally, combined with the golfers income data practical problem are calculated, the results show that the proposed method is feasible and convenient for application.     

Risk performance of a pre-test estimator for normal variance with the Stein-variance estimator under the LINEX loss function

In this paper, we derive the exact formula of the risk function of a pre-test estimator for normal variance with the Stein-variance (PTSV) estimator when the asymmetric LINEX loss function is used. Fixing the critical value of the pre-test to unity which is a suggested critical value in some sense, we examine numerically the risk performance of the PTSV estimator based on the risk function derived. Our numerical results show that although the PTSV estimator does not dominate the usual variance estimator when under-estimation is more severe than over-estimation, the PTSV estimator dominates the usual variance estimator when over-estimation is more severe. It is also shown that the dominance of the PTSV estimator over the original Stein-variance estimator is robust to the extension from the quadratic loss function to the LINEX loss function.     

Bayes sequential estimation for a Poisson process under a LINEX loss function

In this paper, within the framework of a Bayesian model, we consider the problem of sequentially estimating the intensity parameter of a homogeneous Poisson process with a linear exponential (LINEX) loss function and a fixed cost per unit time. An asymptotically pointwise optimal (APO) rule is proposed. It is shown to be asymptotically optimal for the arbitrary priors and asymptotically non-deficient for the conjugate priors in a similar sense of Bickel and Yahav [Asymptotically pointwise optimal procedures in sequential analysis, in Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, University of California Press, Berkeley, CA, 1967, pp. 401–413; Asymptotically optimal Bayes and minimax procedures in sequential estimation, Ann. Math. Statist. 39 (1968), pp. 442–456] and Woodroofe [A.P.O. rules are asymptotically non-deficient for estimation with squared error loss, Z. Wahrsch. verw. Gebiete 58 (1981), pp. 331–341], respectively. The proposed APO rule is illustrated using a real data set.     

Admissibility of estimators in the non-regular family under entropy loss function

In this paper we have considered the problem of finding admissible estimates for a fairly general class of parametric functions in the so called “non-regular” type of densities. The admissibility of generalized Bayes and Pitman estimates of functions of parameters have been established under entropy loss function.     

Estimation of the location parameter and the average worth of the selected subset of two parameter exponential populations under LINEX loss function

Suppose a subset of populations is selected from k exponential populations with unknown location parameters θ₁, θ₂, …, θ_k and common known scale parameter σ. We consider the estimation of the location parameter of the selected population and the average worth of the selected subset under an asymmetric LINEX loss function. We show that the natural estimator of these parameters is biased and find the uniformly minimum risk-unbiased (UMRU) estimator of these parameters. In the case of k = 2, we find the minimax estimator of the location parameter of the smallest selected population. Furthermore, we compare numerically the risk of UMRU, minimax, and the natural estimators.     

A note on the admissibility of relative to the linex loss function

The purpose of this note is to give a correct proof of a result in Rojo (1987). Let 2 be the mean of a random sample of size n from a normal 2 distribution with unknown mean 0 and known variance o . Following earlier work by Zellner (1986), Rojo (1987) considered the admissibility of the linear estimator c; + d relative to Variants (1975) asymmetric LINEX loss function     

Minimax estimation of a probability of success under LINEX loss

Minimax estimation of a binomial probability under LINEX loss function is considered. It is shown that no equalizer estimator is available in the statistical decision problem under consideration. It is pointed out that the problem can be solved by determining the Bayes estimator with respect to a least favorable distribution having finite support. In this situation, the optimal estimator and the least favorable distribution can be determined only by using numerical methods. Some properties of the minimax estimators and the corresponding least favorable prior distributions are provided depending on the parameters of the loss function. The properties presented are exploited in computing the minimax estimators and the least favorable distributions. The results obtained can be applied to determine minimax estimators of a cumulative distribution function and minimax estimators of a survival function.     

Asymptotic Optimal Estimation of Poisson Mean Under LINEX Loss Function

In this article, we consider the problem of sequentially estimating the mean of a Poisson distribution under LINEX (linear exponential) loss function and fixed cost per observation within a Bayesian framework. An asymptotically pointwise optimal rule with a prior distribution is proposed and shown to be asymptotically optimal for arbitrary priors. The proposed asymptotically pointwise optimal rule is illustrated using a real data set.     

Minimax Estimation of the Bounded Parameter of Some Discrete Distributions Under LINEX Loss Function

For a class of discrete distributions, including Poisson(θ), Generalized Poisson(θ), Borel(m, θ), etc., we consider minimax estimation of the parameter θ under the assumption it lies in a bounded interval of the form [0, m] and a LINEX loss function. Explicit conditions for the minimax estimator to be Bayes with respect to a boundary supported prior are given. Also for Bernoulli(θ)-distribution, which is not in the mentioned class of discrete distributions, we give conditions for which the Bayes estimator of θ ∈ [0, m], m < 1 with respect to a boundary supported prior is minimax under LINEX loss function. Numerical values are given for the largest values of m for which the corresponding Bayes estimators of θ are minimax.     

On the Admissibility of Linear Estimators in a Multivariate Normal Distribution Under LINEX Loss Function

Consider an estimation problem of a linear combination of population means in a multivariate normal distribution under LINEX loss function. Necessary and sufficient conditions for linear estimators to be admissible are given. Further, it is shown that the result is an extension of the quadratic loss case as well as the univariate normal case.     

ESTIMATION OF THE PARAMETER OF A POISSON DISTRIBUTION USING A LINEX LOSS FUNCTION

This paper considers estimation of the parameter of a Poisson distribution using Varian's (1975) asymmetric LINEX loss function L (δ) = b{exp(aδ) - aδ - 1}, where δ is the estimation error and b > 0, a 0. It is shown that for a < 0, the sample mean X¯ is admissible whereas for a > 0, X¯ is dominated by c*X¯, where c*= (n/a)log(1+a/n). Practical implications of this result are indicated. More general results, concerning the admissibility of estimators of the form cX¯+ d are also presented.     

Admissibility of generalized bayes and pitman estimates in the non-regular family

In this paper we have considered the problem of finding admissible estimates for a fairly general class of parametric functions in the so called “non-regular” type of densities Following Karlin s (1958) technique, we have established the ad-missibility of generalized Bayes estimates and Pitman estimates. Some examples are discussed.     

Shrinkage estimator of regression model under asymmetric loss

This article investigates the performance of the shrinkage estimator (SE) of the parameters of a simple linear regression model under the LINEX loss criterion. The risk function of the estimator under the asymmetric LINEX loss is derived and analyzed. The moment-generating functions and the first two moments of the estimators are also obtained. The risks of the SE have been compared numerically with that of pre-test and least-square estimators (LSEs) under the LINEX loss criterion. The numerical comparison reveals that under certain conditions the LSE is inadmissible, and the SE is the best among the three estimators.     

Preliminary test and Bayes Estimation of a Location Parameter Under Blinex Loss

In this article, the preliminary test estimator is considered under the BLINEX loss function. The problem under consideration is the estimation of the location parameter from a normal distribution. The risk under the null hypothesis for the preliminary test estimator, the exact risk function for restricted maximum likelihood and approximated risk function for the unrestricted maximum likelihood estimator, are derived under BLINEX loss and the different risk structures are compared to one another both analytically and computationally. As a motivation on the use of BLINEX rather than LINEX, the risk for the preliminary test estimator under BLINEX loss is compared to the risk of the preliminary test estimator under LINEX loss and it is shown that the LINEX expected loss is higher than BLINEX expected loss. Furthermore, two feasible Bayes estimators are derived under BLINEX loss, and a feasible Bayes preliminary test estimator is defined and compared to the classical preliminary test estimator.     

Bayes Estimation Based on Joint Progressive Type II Censored Data Under LINEX Loss Function

In the lifetime experiments, the joint censoring scheme is useful for planning comparative purposes of two identical products manufactured coming from different lines. In this article, we will confine ourselves to the data obtained by conducting a joint progressive Type II censoring scheme on the basis of the two combined samples selected from the two lines. Moreover, it is supposed that the distributions of lifetimes of the two products satisfy in a proportional hazard model. A general form for the distributions is considered, and we tackle the problem of obtaining Bayes estimates under the squared error and linear-exponential (LINEX) loss functions. As a special case, the Weibull distribution is discussed in more detail. Finally, the estimated risks of the various estimators obtained are compared using the Monte Carlo method.