Exact linear inference and prediction for exponential distributions based on general progressively type-ii censored samples

In this paper, we make use of an algorithm of Huffer and Lin (2001) in order to develop exact interval estimation for the location and scale parameters of an exponential distribution based on general progressively Type-II censored samples. The exact prediction intervals for failure times of the items censored at the last observation are also presented for one-parameter and two-parameter exponential distributions. Finally, we give two examples to illustrate the methods of inference developed here.     

On the Bayesian inference of Kumaraswamy distributions based on censored samples

In this article we discuss Bayesian estimation of Kumaraswamy distributions based on three different types of censored samples. We obtain Bayes estimates of the model parameters using two different types of loss functions (LINEX and Quadratic) under each censoring scheme (left censoring, singly type-II censoring, and doubly type-II censoring) using Monte Carlo simulation study with posterior risk plots for each different choices of the model parameters. Also, detailed discussion regarding elicitation of the hyperparameters under the dependent prior setup is discussed. If one of the shape parameters is known then closed form expressions of the Bayes estimates corresponding to posterior risk under both the loss functions are available. To provide the efficacy of the proposed method, a simulation study is conducted and the performance of the estimation is quite interesting. For illustrative purpose, real-life data are considered.     

Correcting moments for goodness of fit tests based on two entropy estimates

The sample entropy (Vasicek, 1976) has been most widely used as a nonparametric entropy estimator due to its simplicity, but its underlying distribution function has not been known yet though its moments are required in establishing the entropy-based goodness of test statistic (Soofi et al., 1995). In this paper we derive the nonparametric distribution function of the sample entropy as a piece-wise uniform distribution in the lights of Theil (1980) and Dudwicz and van der Meulen (1987). Then we establish the entropy-based goodness of fit test statistics based on the nonparametric distribution functions of the sample entropy and modified sample entropy (Ebrahimi et al., 1994), and compare their performances for the exponential and normal distributions.     

Exact nonparametric confidence,prediction and tolerance intervals based on multi-sample Type-II right censored data

Exact nonparametric inference based on ordinary Type-II right censored samples has been extended here to the situation when there are multiple samples with Type-II censoring from a common continuous distribution. It is shown that marginally, the order statistics from the pooled sample are mixtures of the usual order statistics with multivariate hypergeometric weights. Relevant formulas are then derived for the construction of nonparametric confidence intervals for population quantiles, prediction intervals, and tolerance intervals in terms of these pooled order statistics. It is also shown that this pooled-sample approach assists in achieving higher confidence levels when estimating large quantiles as compared to a single Type-II censored sample with same number of observations from a sample of comparable size. We also present some examples to illustrate all the methods of inference developed here.     

Consistency and robustness of tests and estimators based on depth

In this paper it is shown that data depth does not only provide consistent and robust estimators but also consistent and robust tests. Thereby, consistency of a test means that the Type I (α$\alpha$) error and the Type II (β$\beta$) error converge to zero with growing sample size in the interior of the nullhypothesis and the alternative, respectively. Robustness is measured by the breakdown point which depends here on a so-called concentration parameter. The consistency and robustness properties are shown for cases where the parameter of maximum depth is a biased estimator and has to be corrected. This bias is a disadvantage for estimation but an advantage for testing. It causes that the corresponding simplicial depth is not a degenerated U-statistic so that tests can be derived easily. However, the straightforward tests have a very poor power although they are asymptotic α-level$\alpha \mathrm{-}\mathrm{level}$ tests. To improve the power, a new method is presented to modify these tests so that even consistency of the modified tests is achieved. Examples of two-dimensional copulas and the Weibull distribution show the applicability of the new method.     

Some robust two-sample test statistics based on m-estimators of location

This paper describes a simulation experiment that compares the performance, in terms of the size and a function of the power, of four two-sample test statistics based on M-estimators for location. M-esti-mates are chosen to ensure similar levels of breakdown point, gross error sensitivity and as far as possible, similar rejection point. Two pairs of sample size and six different distributions are involved. Matching 97.5% critical values for the statistics are determined.     

Preliminary test Liu-type estimators based on W,LR, and LM test statistics in a regression model

This article discusses the preliminary test approach for the regression parameter in multiple regression model. The preliminary test Liu-type estimators based on the Wald (W), Likelihood ratio (LR), and Lagrangian multiplier(LM) tests are presented, when it is supposed that the regression parameter may be restricted to a subspace. We also give the bias and mean squared error of the proposed estimators and the superior of the proposed estimators is also discussed.     

Outlier-detection tests and robust estimators based on signs of residuals

The asymptotic structure of a vector of weighted sums of signs of residuals, in the general linear model, is studied. The vector can be used as a basis for outlier-detection tests, or alternatively, setting the vector to zero and solving for the parameter yields a class of robust estimators which are analogues of the sample median. Asymptotic results for both estimates and tests are obtained. The question of optimal weights is investigated, and the optimal estimators in the case of simple linear regression are found to coincide with estimators introduced by Adichie.     

On the equivalence of some test criteria based on BAN estimators for the multivariate exponential family

For the general multivariate exponential family of distributions it is shown that Rao's test criterion based on efficient scores is algebraically identical to the general chi-squared criterion based on maximum likelihood estimates and, similarly, that the Wald statistic is algebraically identical to the general minimum modified chi-squared statistic using linearization; these results are valid also for the multisample versions. Thus, these are extensions to the general exponential family of the findings due to Silvey (1970) and Bhapkar (1966), respectively, for the special case of the multinomial family.It is also shown that the general forms of the chi-squared and modified chi-squared criteria reduce to their respective well-known forms for the multivariate symmetric power series distribution. This finding is, thus, an extension of results noted by Ferguson (1958) and Clickner (1976) for the special case of the multinomial distribution.     

Parameter and quantile estimation for the three-parameter gamma distribution based on statistics invariant to unknown location

The three-parameter gamma distribution is widely used as a model for distributions of life spans, reaction times, and for other types of skewed data. In this paper, we propose an efficient method of estimation for the parameters and quantiles of the three-parameter gamma distribution, which avoids the problem of unbounded likelihood, based on statistics invariant to unknown location. Through a Monte Carlo simulation study, we then show that the proposed method performs well compared to other prominent methods in terms of bias and mean squared error. Finally, we present two illustrative examples.     

Performance of the shrinkage preliminary test ridge regression estimators based on the conflicting of W,LR and LM tests

The shrinkage preliminary test ridge regression estimators (SPTRRE) based on the Wald (W), the likelihood ratio (LR) and the Lagrangian multiplier (LM) tests are considered in this paper. The bias and the risk functions of the proposed estimators are derived. The regions of optimality of the estimators are determined under the quadratic risk function. Under the null hypothesis, the SPTRRE based on LM test has the smallest risk, followed by the estimators based on LR and W tests. However, the SPTRRE based on W test performs the best followed by the LR and LM based estimators when the parameter moves away from the subspace of the restrictions. The conditions of superiority of the proposed estimator for both ridge and departure parameters are discussed. The optimum choice of the level of significance becomes the traditional choice by using the W test for all non-negative ridge parameters.     

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.     

Comparison of preliminary test estimators based on generalized order statistics from proportional hazard family using Pitman measure of closeness

In this article, based on generalized order statistics from a family of proportional hazard rate model, we use a statistical test to generate a class of preliminary test estimators and shrinkage preliminary test estimators for the proportionality parameter. These estimators are compared under Pitman measure of closeness (PMC) as well as MSE criteria. Although the PMC suffers from non transitivity, in the first class of estimators, it has the transitivity property and we obtain the Pitman-closest estimator. Analytical and graphical methods are used to show the range of parameter in which preliminary test and shrinkage preliminary test estimators perform better than their competitor estimators. Results reveal that when the prior information is not too far from its real value, the proposed estimators are superior based on both mentioned criteria.     

Some nonparametric precedence-type tests based on progressively censored samples and evaluation of power

In this paper, we consider the problem of testing the equality of two distributions when both samples are progressively Type-II censored. We discuss the following two statistics: one based on the Wilcoxon-type rank-sum precedence test, and the second based on the Kaplan–Meier estimator of the cumulative distribution function. The exact null distributions of these test statistics are derived and are then used to generate critical values and the corresponding exact levels of significance for different combinations of sample sizes and progressive censoring schemes. We also discuss their non-null distributions under Lehmann alternatives. A power study of the proposed tests is carried out under Lehmann alternatives as well as under location-shift alternatives through Monte Carlo simulations. Through this power study, it is shown that the Wilcoxon-type rank-sum precedence test performs the best.     

On the likelihood estimation of the parameters of Gompertz distribution based on complete and progressively Type-II censored samples

In this paper, by considering a progressively Type-II censored sample from the two-parameter Gompertz distribution, a necessary and sufficient condition is established for the existence and uniqueness of the maximum-likelihood estimates of the shape and scale parameters. The results for the special cases of complete and ordinary Type-II right censored samples are then deduced. Several numerical examples from the literature are presented for the purpose of illustrating the established results.     

Estimation of the location and scale parameters of the extreme value distmbution based on multiply type-II censored samples

In this paper, we consider the problem of estimating the location and scale parameters of an extreme value distribution based on multiply Type-II censored samples. We first describe the best linear unbiased estimators and the maximum likelihood estimators of these parameters. After observing that the best linear unbiased estimators need the construction of some tables for its coefficients and that the maximum likelihood estimators do not exist in an explicit algebraic form and hence need to be found by numerical methods, we develop approximate maximum likelihood estimators by appropriately approximating the likelihood equations. In addition to being simple explicit estimators, these estimators turn out to be nearly as efficient as the best linear unbiased estimators and the maximum likelihood estimators. Next, we derive the asymptotic variances and covariance of these estimators in terms of the first two single moments and the product moments of order statistics from the standard extreme value distribution. Finally, we present an example in order to illustrate all the methods of estimation of parameters discussed in this paper.     

E-Bayesian estimation and its E-posterior risk of the exponential distribution parameter based on complete and type I censored samples

Abstract

This article studies E-Bayesian estimation and its E-posterior risk, for failure rate derived from exponential distribution, in the case of the two hyper parameters. In order to measure the estimated risk, the definition of E-posterior risk (expected posterior risk) is proposed based on the definition of E-Bayesian estimation. Moreover, under the different prior distributions of hyper parameters, the formulas of E-Bayesian estimation and formulas of E-posterior risk are given respectively, these estimations are derived based on a conjugate prior distribution for the unknown parameter under the squared error loss function. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation and a real data set have been analyzed for illustrative purposes, results are compared on the basis of E-posterior risk.