An identity for exponential distributions with the common location parameter and its applications

An identity for exponential distributions with an unknown common location parameter and unknown and possibly unequal scale parameters is established.Through use of the identity the maximum likelihood estimator (MLE) and the uniformly minimum variance unbiased estimator (UMVUE) of a quantile of an exponential population are compared under the squared error loss.A class of estimators dominating both MLE and UMVUE is obtained by using the identity.     

Hypothesis testing on the common location parameter of several shifted exponential distributions: A note

This note deals with hypothesis testing on the common location parameter of several shifted exponential distributions with unknown and possibly unequal scale parameters. No exact test is available for the above mentioned problem; and one does not have the luxury of applying the asymptotic Chi-square test for the likelihood ratio test statistic since the distributions do not satisfy the usual regularity conditions. Therefore, we have proposed a few approximate tests based on the parametric bootstrap method which appear to work well even for small samples in terms of attaining the level. Powers of the proposed tests have been provided along with a recommendation of their usage.     

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.     

Estimation of common location parameter of several heterogeneous exponential populations based on generalized order statistics

In this article, several independent populations following exponential distribution with common location parameter and unknown and unequal scale parameters are considered. From these populations, several independent samples of generalized order statistics (gos) are drawn. Under the setup of gos, the problem of estimation of common location parameter is discussed and various estimators of common location parameter are derived. The authors obtained maximum likelihood estimator (MLE), modified MLE and uniformly minimum variance unbiased estimator of common location parameter. Furthermore, under scaled-squared error loss function, a general inadmissibility result of invariant estimator is proposed. The derived results are further reduced for upper record values which is a special case of gos. Finally, simulation study and real life example are reported to show the performances of various competing estimators in terms of percentage risk improvement.     

Multi-stage estimation of the common location parameter of several exponential distributions

The problem of constructing a confidence interval of ‘preassigned width and coverage probability’ considered by Costanza/ Hamdy and Son(1986) is further analyzed. Several multi-stage estimation procedures [ like, purely sequential, accelerated sequential and three-stage procedures ] are utilized to deal with the same estimation problem. The relative advantages and disadvantages of these procedures are discussed.     

Estimation of location parameters of several exponential distributions

In this paper we address the problem of simultaneous estimation of location parameters of several exponential distributions assuming that the scale parameters are unknown and possibly unequal. From a decision theoretic point of view it is shown that the standard estimators are inadmissible and the improved estimators are obtained when p, the number of populations, is more than one.     

A note on the location parameters of two exponential distributions under order restrictions

The problem of simultaneous estimation of location parameters of two independent exponential distributions is considered when location and/or scale parameters are ordered. We show that the standard estimators in the unrestricted case which use information only from the populations individually can be improved upon when various order restrictions are known to hold. The improved estimators are obtained under the quadratic loss function     

Estimation of common location parameter of two exponential populations based on records

Consider the problem of estimating the common location parameter of two exponential populations using record data when the scale parameters are unknown. We derive the maximum likelihood estimator (MLE), the modified maximum likelihood estimator (MMLE) and the uniformly minimum variance unbiased estimator (UMVUE) of the common location parameter. Further, we derive a general result for inadmissibility of an equivariant estimator under the scaled-squared error loss function. Using this result, we conclude that the MLE and the UMVUE are inadmissible and better estimators are provided. A simulation study is conducted for comparing the performances of various competing estimators.     

Estimating the common hazard rate of several exponential distributions

Abstract

In the present communication, we consider the estimation of the common hazard rate of several exponential distributions with unknown and unequal location parameters with a common scale parameter under a general class of bowl-shaped scale invariant loss functions. We have shown that the best affine equivariant estimator (BAEE) is inadmissible by deriving a non smooth improved estimator. Further, we have obtained a smooth estimator which improves upon the BAEE. As an application, we have obtained explicit expressions of improved estimators for special loss functions. Finally, a simulation study is carried out for numerically comparing the risk performance of various estimators.     

Two stage fixed width confidence intervals for the common location parameter of several exponential distributins

Two stage sampling schemes are introduced for use in estimating the common location parameter (guarantee time) of two or more exponential distributions with a confidence interval of prespecified width whose coverage probability is at least a given nominal value. Exact expressions for all moments of order r ≥ 1 of the associated two stage sample sizes and for the actual coverage probabilities are derived. The performance of the procedures in a variety of two population, moderate fixed sample size cases is examined via numerical studies involving both exact calculations and Monte Carlo simulations. No new tables are needed to implement any of the proposed methods. A modified two stage procedure is recommended for practical use     

Estimating the common hazard rate of two exponential distributions with ordered location parameters

This paper is concerned with estimating the common hazard rate of two exponential distributions with unknown and ordered location parameters under a general class of bowl-shaped scale invariant loss functions. The inadmissibility of the best affine equivariant estimator is established by deriving an improved estimator. Another estimator is obtained which improves upon the best affine equivariant estimator. A class of improving estimators is derived using the integral expression of risk difference approach of Kubokawa [A unified approach to improving equivariant estimators. Ann Statist. 1994;22(1):290–299]. These results are applied to specific loss functions. It is further shown that these estimators can be derived for four important sampling schemes: (i) complete and i.i.d. sample, (ii) record values, (iii) type-II censoring, and (iv) progressive Type-II censoring. A simulation study is carried out for numerically comparing the risk performance of these proposed estimators.     

Improved estimation of common location of two exponential populations with order restricted scale parameters using censored samples

ABSTRACT

Estimation of common location parameter of two exponential populations is considered when the scale parameters are ordered using type-II censored samples. A general inadmissibility result is proved which helps in deriving improved estimators. Further, a class of estimators dominating the MLE has been derived by an application of integrated expression of risk difference (IERD) approach of Kubokawa. A discussion regarding extending the results to a general k( ? 2) populations has been done. Finally, all the proposed estimators are compared through simulation.     

A closed testing procedure for comparing successive exponential populations with respect to location parameter

Consider k independent random samples such that i^th sample is drawn from a two-parameter exponential population with location parameter μ_i and scale parameter θ_i,?i = 1, …, k. For simultaneously testing differences between location parameters of successive exponential populations, closed testing procedures are proposed separately for the following cases (i) when scale parameters are unknown and equal and (ii) when scale parameters are unknown and unequal. Critical constants required for the proposed procedures are obtained numerically and the selected values of the critical constants are tabulated. Simulation study revealed that the proposed procedures has better ability to detect the significant differences and has more power in comparison to exiting procedures. The illustration of the proposed procedures is given using real data.     

Estimating ordered quantiles of two exponential populations with a common minimum guarantee time

The problem of estimating ordered quantiles of two exponential populations is considered, assuming equality of location parameters (minimum guarantee times), using the quadratic loss function. Under order restrictions, we propose new estimators which are the isotonized version of the MLEs, call it, restricted MLE. A sufficient condition for improving equivariant estimators is derived under order restrictions on the quantiles. Consequently, estimators improving upon the old estimators have been derived. A detailed numerical study has been done to evaluate the performance of proposed estimators using the Monte-Carlo simulation method and recommendations have been made for the use of the estimators.     

Testimator of the scale parameter of the exponential distribution using linex loss function

The present paper investigates the properties of a testimator of scale of an exponential distribution under Linex loss function. The risk function of testimator is derived and compared with that of an admissible estimator relative to Linex loss function. The shrinkage testimator is proposed which is the extension of testimator and its properties have been discussed. The level of significance of testimator is decided on the basis of Akaike information criterion following Hirano (1977, 1978). It is found that the testimator and shrinkage testimator dominates the admissible estimator in terms of risk in certain parametric space.     

Estimating quantiles of several normal populations with a common mean

Suppose we have k( ? 2) normal populations with a common mean and possibly different variances. The problem of estimation of quantile of the first population is considered with respect to a quadratic loss function. In this paper, we have generalized the inadmissibility results obtained by Kumar and Tripathy (2011 Kumar, S., Tripathy, M.R. (2011). Estimating quantiles of normal populations with a common mean. Commun. Stat. - Theory Methods 40:2719–2736.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) for k = 2 to a general k( ? 2). Moreover, a massive simulation study has been done in order to numerically compare the risk values of various proposed estimators for the cases k = 3 and k = 4 and recommendations are made for the use of estimators under certain situations.     

Estimation of the common scale of several shifted exponential distributions with unknown locations

We consider the estimation of the common scale parameter of two or more independent shifted exponential distributions with unknown locations. Under a large class of bowl-shaped loss functions, the best location-scale in-variant estimator is shown to be inadmissible. A class of improved estimators is derived. Some numerical results are presented to show the magnitude of risk reduction.