Discriminating between generalized exponential,geometric extreme exponential and Weibull distributions

Generalized exponential, geometric extreme exponential and Weibull distributions are three non-negative skewed distributions that are suitable for analysing lifetime data. We present diagnostic tools based on the likelihood ratio test (LRT) and the minimum Kolmogorov distance (KD) method to discriminate between these models. Probability of correct selection has been calculated for each model and for several combinations of shape parameters and sample sizes using Monte Carlo simulation. Application of LRT and KD discrimination methods to some real data sets has also been studied.     

Exact likelihood inference for two populations from two-parameter exponential distributions under joint Type-II censoring

In this article, we develop exact inference for two populations that have a two-parameter exponential distribution with the same location parameter and different scale parameters when Type-II censoring is implemented on the two samples in a combined manner. We obtain the conditional maximum likelihood estimators (MLEs) of the three parameters. We then derive the exact distributions of these MLEs along with their moment generating functions. Based on general entropy loss function, Bayesian study about the parameters is presented. Finally, some simulation results and an illustrative example are presented to illustrate the methods of inference developed here.     

Testing equality of location parameters of two exponential distributions from censored samples

A modification to Tiku's (1981) test, which may be seriously biased, is proposed. The modified test is only marginally biased if at all and is substantially more powerful. A ratio test based on Tiku’s (1967) modified likelihood function is also proposed, and shown to have power comparable to the power of the ratio test based on the likelihood function. The proposed ratio test is, however, much easier from a computational viewpoint.     

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     

Asymptotic non-null distribution of a test connected with exponential populations

In this paper asymptotic expansions of the non-null distribution of the likelihood ratio criterion for testing the equality of several one parameter exponential distributions are obtained under local alternatives. These expansions are in terms of Chi-square distributions.     

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.     

A small sample test for an absolutely continuous bivariate exponential model

The finite sample distribution of the likelihood ratio sta-tistic is obtained for testing independence, given marginal homo-geneity, in the absolutely continuous bivariate exponential distri-bution of Block and Basu (1974). This test is discussed in light of the analysis of Gross and Lam (1981) on times to relief of head-aches for standard and new treatments on ten subjects.     

Approximate MLE for the scale parameter of the generalized exponential distribution under random censoring

In this paper, we consider the maximum likelihood estimator (MLE) of the scale parameter of the generalized exponential (GE) distribution based on a random censoring model. We assume the censoring distribution also follows a GE distribution. Since the estimator does not provide an explicit solution, we propose a simple method of deriving an explicit estimator by approximating the likelihood function. In order to compare the performance of the estimators, Monte Carlo simulation is conducted. The results show that the MLE and the approximate MLE are almost identical in terms of bias and variance.     

Testing the equality of location parameters of exponential populations from censored samples

The modified Tiku test and the modified likelihood ratio test proposed by Tiku and Vaughan (1991) for k=2 exponential populations are extended to . k > 2 populations. These tests are shown to be more powerful than the test proposed by Kambo and Awad (1985). Unlike the Kambo-Awad test, the proposed tests are shown to have almost symmetric power functions. Further, these tests can be applied when there is both left or right censoring present, in contrast to the tests of Sukhatme (1937), Bain and Englehardt (1991) and Elewa et al. (1992), who assume that there is no left censoring.     

Testtesting for the equality of scale parameters of k(> 2) exponential populations based on complete and type ii censored samples

This paper is concerned with testing the equality of scale parameters of K(> 2) two-parameter exponential distributions in presence of unspecified location parameters based on complete and type II censored samples. We develop a marginal likelihood ratio statistic, a quadratic statistic (Qu) (Nelson, 1982) based on maximum marginal likelihood estimates of the scale parameters under the null and the alternative hypotheses, a C(a) statistic (CPL) (Neyman, 1959) based on the profile likelihood estimate of the scale parameter under the null hypothesis and an extremal scale parameter ratio statistic (ESP) (McCool, 1979). We show that the marginal likelihood ratio statistic is equivalent to the modified Bartlett test statistic. We use Bartlett's small sample correction to the marginal likelihood ratio statistic and call it the modified marginal likelihood ratio statistic (MLB). We then compare the four statistics, MLBi Qut CPL and ESP in terms of size and power by using Monte Carlo simulation experiments. For the variety of sample sizes and censoring combinations and nominal levels considered the statistic MLB holds nominal level most accurately and based on empirically calculated critical values, this statistic performs best or as good as others in most situations. Two examples are given.     

Exact likelihood inference for two exponential populations based on a joint generalized Type-I hybrid censored sample

Following the work of Chen and Bhattacharyya [Exact confidence bounds for an exponential parameter under hybrid censoring. Comm Statist Theory Methods. 1988;17:1857–1870], several results have been developed regarding the exact likelihood inference of exponential parameters based on different forms of censored samples. In this paper, the conditional maximum likelihood estimators (MLEs) of two exponential mean parameters are derived under joint generalized Type-I hybrid censoring on the two samples. The moment generating functions (MGFs) and the exact densities of the conditional MLEs are obtained, using which exact confidence intervals are then developed for the model parameters. We also derive the means, variances, and mean squared errors of these estimates. An efficient computational method is developed based on the joint MGF. Finally, an example is presented to illustrate the methods of inference developed here.     

On a life testing problem involving the two parameter exponential distribution

For a hypothesis testing problem involving the location and scale parameters of an exponential distribution, Perng (1977) proposed a test procedure based on the first r out of n observed failure times. In this paper the likelihood ratio test is determined, critical values are provided and the asymptotic null distribution is determined. An alternate test based on an F statistic is also proposed and the critical regions and power functions of the procedures are compared.     

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.     

Tests for the skewness parameter of two-piece double exponential distribution in the presence of nuisance parameters

In this paper, tests for the skewness parameter of the two-piece double exponential distribution are derived when the location parameter is unknown. Classical tests like Neyman structure test and likelihood ratio test (LRT), that are generally used to test hypotheses in the presence of nuisance parameters, are not feasible for this distribution since the exact distributions of the test statistics become very complicated. As an alternative, we identify a set of statistics that are ancillary for the location parameter. When the scale parameter is known, Neyman–Pearson's lemma is used, and when the scale parameter is unknown, the LRT is applied to the joint density function of ancillary statistics, in order to obtain a test for the skewness parameter of the distribution. Test for symmetry of the distribution can be deduced as a special case. It is found that power of the proposed tests for symmetry is only marginally less than the power of corresponding classical optimum tests when the location parameter is known, especially for moderate and large sample sizes.     

A pseudo restricted MLE under multivariate order restrictions and its algorithm

Abstract

It is shown in this paper that a quasi order for the vectors in R^p is a cone induced if and only if the order is preservable under limits and under linear combinations with non-negative coefficients. For the mean vectors in MANOVA subject to the restriction of simple ordering, a pseudo restricted MLE is proposed. This estimator is a matrix projection onto a closed convex set inside the restricted domain. An algorithm for the pseudo restricted MLE is developed, that computes the matrix projections using only vector projections.     

Simultaneous inferences for ordered exponential location parameters under unbalanced data and heteroscedasticity of scale parameters

A test procedure for testing homogeneity of location parameters against simple ordered alternative is proposed for k(k ≥ 2) members of two parameter exponential distribution under unbalanced data and heteroscedasticity of the scale parameters. The relevant one-sided and two-sided simultaneous confidence intervals (SCIs) for all k(k ? 1)/2 ordered pairwise differences of location parameters are also proposed. Simulation-based study revealed that the proposed procedure is better than the recently proposed procedure in terms of power, coverage probability, and average volume of SCIs. The implementation of proposed procedure is demonstrated through real life data.     

Change-point analysis with bathtub shape for the exponential distribution

Likelihood ratio type test statistic and Schwarz information criterion statistics are proposed for detecting possible bathtub-shaped changes in the parameter in a sequence of exponential distributions. The asymptotic distribution of likelihood ratio type statistic under the null hypothesis and the testing procedure based on Schwarz information criterion are derived. Numerical critical values and powers of two methods are tabulated for certain selected values of the parameters. The tests are applied to detect the change points for the predator data and Stanford heart transplant data.     

Bayesian inference based on a jointly type-II censored sample from two exponential populations

In this paper, based on a jointly type-II censored sample from two exponential populations, the Bayesian inference for the two unknown parameters are developed with the use of squared-error, linear-exponential and general entropy loss functions. The problem of predicting the future failure times, both point and interval prediction, based on the observed joint type-II censored data, is also addressed from a Bayesian viewpoint. A Monte Carlo simulation study is conducted to compare the Bayesian estimators with the maximum likelihood estimator developed by Balakrishnan and Rasouli [Exact likelihood inference for two exponential populations under joint type-II censoring. Comput Stat Data Anal. 2008;52:2725–2738]. Finally, a numerical example is utilized for the purpose of illustration.