On the power of goodness-of-fit tests for the exponential distribution under progressive Type-II censoring

The aim of this article is to review existing goodness-of-fit tests for the exponential distribution under progressive Type-II censoring and to provide some new ideas and adjustments. In particular, we consider two-parameter exponentially distributed random variables and adapt the proposed test procedures to our scenario if necessary. Then, we compare their power by an extensive simulation study. Furthermore, we propose five new test procedures that provide reasonable alternatives to those already known.     

A new two sample type-II progressive censoring scheme

In this paper we introduce a new type-II progressive censoring scheme for two samples. It is observed that the proposed censoring scheme is analytically more tractable than the existing joint progressive type-II censoring scheme proposed by Rasouli and Balakrishnan. The maximum likelihood estimators of the unknown parameters are obtained and their exact distributions are derived. Based on the exact distributions of the maximum likelihood estimators exact confidence intervals are also constructed. For comparison purposes we have used bootstrap confidence intervals also. One data analysis has been performed for illustrative purposes. Finally we propose some open problems.     

Exact inference on multiple exponential populations under a joint type-II progressive censoring scheme

ABSTRACT

Recently Mondal and Kundu [Mondal S, Kundu D. A new two sample type-II progressive censoring scheme. Commun Stat Theory Methods. 2018. doi:10.1080/03610926.2018.1472781] introduced a Type-II progressive censoring scheme for two populations. In this article, we extend the above scheme for more than two populations. The aim of this paper is to study the statistical inference under the multi-sample Type-II progressive censoring scheme, when the underlying distributions are exponential. We derive the maximum likelihood estimators (MLEs) of the unknown parameters when they exist and find out their exact distributions. The stochastic monotonicity of the MLEs has been established and this property can be used to construct exact confidence intervals of the parameters via pivoting the cumulative distribution functions of the MLEs. The distributional properties of the ordered failure times are also obtained. The Bayesian analysis of the unknown model parameters has been provided. The performances of the different methods have been examined by extensive Monte Carlo simulations. We analyse two data sets for illustrative purposes.     

Two-sample Pitman closeness comparison under progressive Type-II censoring

In this paper, we consider two problems concerning two independent progressively Type-II censored samples. We first consider the Pitman closeness (PC) of order statistics from two independent progressively censored samples to a specific population quantile. We then consider the point prediction of a future progressively censored order statistic and discuss the determination of the closest progressively censored order statistic from the current sample according to the simultaneous closeness probabilities. For both these problems, explicit expressions are derived for the pertinent PC probabilities, and then special cases are given as examples. For various censoring schemes, we also present numerical results for the standard uniform, standard exponential, and standard normal distributions. Finally, a distribution-free result for the median is obtained.     

Robust Bayesian analysis for exponential parameters under generalized Type-II progressive hybrid censoring

The Type-II progressive hybrid censoring scheme has received wide attention, but it has a disadvantage in that long time may be required to complete the life test. The generalized progressive Type-II hybrid censoring scheme has recently been proposed to solve this problem. Under the censoring scheme, the time on test does not exceed a predetermined time. In this paper, we propose a robust Bayesian approach based on a hierarchical structure when the generalized progressive Type-II hybrid censored sample has a two-parameter exponential distribution. For unknown parameters, marginal posterior distributions are provided in closed forms, and their statistical properties are discussed. To examine the robustness of the proposed method, Monte Carlo simulations are conducted and a real data set is analyzed. Further, the quality and adequacy of the proposed model are evaluated in an analysis based on the real data.     

Estimation of parameters of Kumaraswamy-Exponential distribution under progressive type-II censoring

In this paper, the problem of estimating unknown parameters of a two-parameter Kumaraswamy-Exponential (Kw-E) distribution is considered based on progressively type-II censored sample. The maximum likelihood (ML) estimators of the parameters are obtained. Bayes estimates are also obtained using different loss functions such as squared error, LINEX and general entropy. Lindley's approximation method is used to evaluate these Bayes estimates. Monte Carlo simulation is used for numerical comparison between various estimates developed in this paper.     

Comparison of two life distributions on the basis of their percentile residual life functions

Let F and G be life distributions with respective failure rate functions r_F and r_G and respective 100α-percentile (0 < α < 1) residual life functions q_{α, F}, and q_{α, G}. Distribution-free two-sample tests are proposed for testing H₀: F = G against H_1,α,: q_{α, F} ≥ q_{α, G} and H_{2 α}: q_{β, F} ≥ q_β,G for all β ≥ α. This class of tests includes as a special case the test of Kochar (1981) for the alternative H*₂: r_F ≤ r_G. A theorem of Govindarajulu (1976) is extended in order to obtain asymptotic normality of the test statistics. The condition q_{α, F} ≥ q_{α, G} is implied by r_F ≤ r_G and is unrelated to the stochastic ordering F≤ G; if F and G are Weibull distributions with respective shape parameters c₁ and c₂ such that 1 ≤ C₁ < C₂, then q_α,F ≥ q_{α, G} for all α larger than a function of the parameters.     

Bias corrected MLEs under progressive type-II censoring scheme

ABSTRACT

Censoring frequently occurs in survival analysis but naturally observed lifetimes are not of a large size. Thus, inferences based on the popular maximum likelihood (ML) estimation which often give biased estimates should be corrected in the sense of bias. Here, we investigate the biases of ML estimates under the progressive type-II censoring scheme (pIIcs). We use a method proposed in Efron and Johnstone [Fisher's information in terms of the hazard rate. Technical Report No. 264, January 1987, Stanford University, Stanford, California; 1987] to derive general expressions for bias corrected ML estimates under the pIIcs. This requires derivation of the Fisher information matrix under the pIIcs. As an application, exact expressions are given for bias corrected ML estimates of the Weibull distribution under the pIIcs. The performance of the bias corrected ML estimates and ML estimates are compared by simulations and a real data application.     

Robust Bayesian estimation of a bathtub-shaped distribution under progressive Type-II censoring

The bathtub-shaped failure rate function has been used for modeling the life spans of a number of electronic and mechanical products, as well as for modeling the life spans of humans, especially when some of the data are censored. This article addresses robust methods for the estimation of unknown parameters in a two-parameter distribution with a bathtub-shaped failure rate function based on progressive Type-II censored samples. Here, a class of flexible priors is considered by using the hierarchical structure of a conjugate prior distribution, and corresponding posterior distributions are obtained in a closed-form. Then, based on the square error loss function, Bayes estimators of unknown parameters are derived, which depend on hyperparameters as parameters of the conjugate prior. In order to eliminate the hyperparameters, hierarchical Bayesian estimation methods are proposed, and these proposed estimators are compared to one another based on the mean squared error, through Monte Carlo simulations for various progressively Type-II censoring schemes. Finally, a real dataset is presented for the purpose of illustration.     

Fisher information in progressive hybrid censoring schemes

The hybrid censoring scheme, which is a mixture of Type-I and Type-II censoring schemes, has been extended to the case of progressive censoring schemes by Kundu and Joarder [Analysis of Type-II progressively hybrid censored data, Comput. Stat. Data Anal. 50 (2006), pp. 2509–2528] and Childs et al. [Exact likelihood inference for an exponential parameter under progressive hybrid censoring schemes, in Statistical Models and Methods for Biomedical and Technical Systems, F. Vonta, M. Nikulin, N. Limnios, and C. Huber-Carol, eds., Birkhäuser, Boston, MA, 2007, pp. 323–334]. In this paper, we derive a simple expression for the Fisher information contained in Type-I and Type-II progressively hybrid censored data. An illustrative example is provided applicable to a scaled-exponential distribution to demonstrate our methodologies.     

Estimating the Pareto parameters under progressive censoring data for constant-partially accelerated life tests

Accelerated life testing is widely used in product life testing experiments since it provides significant reduction in time and cost of testing. In this paper, assuming that the lifetime of items under use condition follow the two-parameter Pareto distribution of the second kind, partially accelerated life tests based on progressively Type-II censored samples are considered. The likelihood equations of the model parameters and the acceleration factor are reduced to a single nonlinear equation to be solved numerically to obtain the maximum-likelihood estimates (MLEs). Based on normal approximation to the asymptotic distribution of MLEs, the approximate confidence intervals (ACIs) for the parameters are derived. Two bootstrap CIs are also proposed. The classical Bayes estimates cannot be obtained in explicit form, so we propose to apply Markov chain Monte Carlo method to tackle this problem, which allows us to construct the credible interval of the involved parameters. Analysis of a simulated data set has also been presented for illustrative purposes. Finally, a Monte Carlo simulation study is carried out to investigate the precision of the Bayes estimates with MLEs and to compare the performance of different corresponding CIs considered.     

A- and D-optimal progressive Type-II censoring designs based on Fisher information

Fisher information about multiple parameters in a progressively Type-II censored sample is discussed. A representation of the Fisher information matrix in terms of the hazard rate of the baseline distribution is established which can be used for efficient computation of the Fisher information. This expression generalizes a result of Zheng and Park [On the Fisher information in multiply censored and progressively censored data, Comm. Statist. Theory Methods 33 (2004), pp. 1821–1835] for Fisher information about a single parameter. The result is applied to identify A- and D-optimal censoring plans in a progressively Type-II censored experiment. For illustration, extreme value, normal, and Lomax distributions are considered.     

Estimating the parameters of a bathtub-shaped distribution under progressive type-II censoring

We consider the problem of estimating unknown parameters, reliability function and hazard function of a two parameter bathtub-shaped distribution on the basis of progressive type-II censored sample. The maximum likelihood estimators and Bayes estimators are derived for two unknown parameters, reliability function and hazard function. The Bayes estimators are obtained against squared error, LINEX and entropy loss functions. Also, using the Lindley approximation method we have obtained approximate Bayes estimators against these loss functions. Some numerical comparisons are made among various proposed estimators in terms of their mean square error values and some specific recommendations are given. Finally, two data sets are analyzed to illustrate the proposed methods.     

Estimation for the three-parameter inverse Gaussian distribution under progressive Type-II censoring

Inverse Gaussian distribution has been used widely as a model in analysing lifetime data. In this regard, estimation of parameters of two-parameter (IG2) and three-parameter inverse Gaussian (IG3) distributions based on complete and censored samples has been discussed in the literature. In this paper, we develop estimation methods based on progressively Type-II censored samples from IG3 distribution. In particular, we use the EM-algorithm, as well as some other numerical methods for determining the maximum-likelihood estimates (MLEs) of the parameters. The asymptotic variances and covariances of the MLEs from the EM-algorithm are derived by using the missing information principle. We also consider some simplified alternative estimators. The inferential methods developed are then illustrated with some numerical examples. We also discuss the interval estimation of the parameters based on the large-sample theory and examine the true coverage probabilities of these confidence intervals in case of small samples by means of Monte Carlo simulations.     

Inference and optimal design of multiple constant-stress testing for generalized half-normal distribution under type-II progressive censoring

The generalized half-normal (GHN) distribution and progressive type-II censoring are considered in this article for studying some statistical inferences of constant-stress accelerated life testing. The EM algorithm is considered to calculate the maximum likelihood estimates. Fisher information matrix is formed depending on the missing information law and it is utilized for structuring the asymptomatic confidence intervals. Further, interval estimation is discussed through bootstrap intervals. The Tierney and Kadane method, importance sampling procedure and Metropolis-Hastings algorithm are utilized to compute Bayesian estimates. Furthermore, predictive estimates for censored data and the related prediction intervals are obtained. We consider three optimality criteria to find out the optimal stress level. A real data set is used to illustrate the importance of GHN distribution as an alternative lifetime model for well-known distributions. Finally, a simulation study is provided with discussion.     

Robust Bayesian estimation of a two-parameter exponential distribution under generalized Type-I progressive hybrid censoring

A generalized Type-I progressive hybrid censoring scheme was proposed recently to overcome the limitations of the progressive hybrid censoring scheme. In this article, we provide a robust Bayesian method to estimate the unknown parameters of the two-parameter exponential distribution of a generalized Type-I progressive hybrid censored sample. For each parameter, we derive the marginal posterior density functions and the corresponding Bayesian estimators under the squared error loss function. To assess the proposed method, Monte Carlo simulations are performed using a real dataset.     

Estimation of the two-parameter bathtub-shaped lifetime distribution with progressive censoring

In this paper, we investigate the estimation problem concerning a progressively type-II censored sample from the two-parameter bathtub-shaped lifetime distribution. We use the maximum likelihood method to obtain the point estimators of the parameters. We also provide a method for constructing an exact confidence interval and an exact joint confidence region for the parameters. Two numerical examples are presented to illustrate the method of inference developed here. Finally, Monte Carlo simulation studies are used to assess the performance of our proposed method.     

On the exact distribution of the MLEs based on Type-II progressively hybrid censored data from exponential distributions

ABSTRACT

Distributions of the maximum likelihood estimators (MLEs) in Type-II (progressive) hybrid censoring based on two-parameter exponential distributions have been obtained using a moment generating function approach. Although resulting in explicit expressions, the representations are complicated alternating sums. Using the spacings-based approach of Cramer and Balakrishnan [On some exact distributional results based on Type-I progressively hybrid censored data from exponential distributions. Statist Methodol. 2013;10:128–150], we derive simple expressions for the exact density and distribution functions of the MLEs in terms of B-spline functions. These representations can be easily implemented on a computer and provide an efficient method to compute density and distribution functions as well as moments of Type-II (progressively) hybrid censored order statistics.