Exact Non-Parametric Confidence, Prediction and Tolerance Intervals with Progressive Type-II Censoring

Abstract.  This article extends recent results [Scand. J. Statist. 28 (2001) 699] about exact non-parametric inferences based on order statistics with progressive type-II censoring. The extension lies in that non-parametric inferences are now covered where the dependence between involved order statistics cannot be circumvented. These inferences include: (a) tolerance intervals containing at least a specified proportion of the parent distribution, (b) prediction intervals containing at least a specified number of observations in a future sample, and (c) outer and/or inner confidence intervals for a quantile interval of the parent distribution. The inferences are valid for any parent distribution with continuous distribution function. The key result shows how the probability of an event involving k dependent order statistics that are observable/uncensored with progressive type-II censoring can be represented as a mixture with known weights of corresponding probabilities involving k dependent ordinary order statistics. Further applications/developments concerning exact Kolmogorov-type confidence regions are indicated.     

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.     

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.     

On the estimation of the extreme value and normal distribution parameters based on progressive type-II hybrid-censored data

A progressive hybrid censoring scheme is a mixture of type-I and type-II progressive censoring schemes. In this paper, we mainly consider the analysis of progressive type-II hybrid-censored data when the lifetime distribution of the individual item is the normal and extreme value distributions. Since the maximum likelihood estimators (MLEs) of these parameters cannot be obtained in the closed form, we propose to use the expectation and maximization (EM) algorithm to compute the MLEs. Also, the Newton–Raphson method is used to estimate the model parameters. The asymptotic variance–covariance matrix of the MLEs under EM framework is obtained by Fisher information matrix using the missing information and asymptotic confidence intervals for the parameters are then constructed. This study will end up with comparing the two methods of estimation and the asymptotic confidence intervals of coverage probabilities corresponding to the missing information principle and the observed information matrix through a simulation study, illustrated examples and real data analysis.     

Prediction intervals for ordinary and dual generalized order statistics from two independent sequences

ABSTRACT

Based on the observed dual generalized order statistics drawn from an arbitrary unknown distribution, nonparametric two-sided prediction intervals as well as prediction upper and lower bounds for an ordinary and a dual generalized order statistic from another iid sequence with the same distribution are developed. The prediction intervals for dual generalized order statistics based on the observed ordinary generalized order statistics are also developed. The coverage probabilities of these prediction intervals are exact and free of the parent distribution, F. Finally, numerical computations and real examples of the coverage probabilities are presented for choosing the appropriate limits of the prediction.     

Optimal Progressive Type-II Censoring Schemes for Nonparametric Confidence Intervals of Quantiles

In this article, optimal progressive censoring schemes are examined for the nonparametric confidence intervals of population quantiles. The results obtained can be universally applied to any continuous probability distribution. By using the interval mass as an optimality criterion, the optimization process is free of the actual observed values from the sample and needs only the initial sample size n and the number of complete failures m. Using several sample sizes combined with various degrees of censoring, the results of the optimization are presented here for the population median at selected levels of confidence (99, 95, and 90%). With the optimality criterion under consideration, the efficiencies of the worst progressive Type-II censoring scheme and ordinary Type-II censoring scheme are also examined in comparison to the best censoring scheme obtained for fixed n and m.     

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.     

Gap-ratio goodness of fit tests for weibull or extreme value distribution assumptions with left or right censored data

With data collection in environmental science and bioassay, left censoring because of nondetects is a problem. Similarly in reliability and life data analysis right censoring frequently occurs. There is a need for goodness of fit tests that can adapt to left or right censored data and be used to check important distributional assumptions without becoming too difficult to regularly implement in practice. A new test statistic is derived from a plot of the standardized spacings between the order statistics versus their ranks. Any linear or curvilinear pattern is evidence against the null distribution. When testing the Weibull or extreme value null hypothesis this statistic has a null distribution that is approximately F for most combinations of sample size and censoring of practical interest. Our statistic is compared to the Mann-Scheuer-Fertig statistic which also uses the standardized spacings between the order statistics. The results of a simulation study show the two tests are competitive in terms of power. Although the Mann-Scheuer-Fertig statistic is somewhat easier to compute, our test enjoys advantages in the accuracy of the F approximation and the availability of a graphical diagnostic.     

Optimal Bayesian variable sampling plans for exponential distributions based on modified type-II hybrid censored samples

For the conventional type-II hybrid censoring scheme (HCS) in Childs et al., a Bayesian variable sampling plan among the class of the maximum likelihood estimators was derived by Lin et al. under the loss function, which does not include the cost of experimental time. Instead of taking the conventional type-II hybrid censoring scheme, a persuasive argument leads to taking the modified type-II hybrid censoring scheme (MHCS) if the cost of experimental time is included in the loss function. In this article, we apply the decision-theoretic approach for the concerned acceptance sampling. With the type-II MHCS, based on a sufficient statistics, the optimal Bayesian sampling plan is derived under a general loss function. Furthermore, for the conjugate prior distribution, the closed-form formula of the Bayes decision rule can be obtained under the quadratic decision loss. Numerical study is given to demonstrate the performance of the proposed Bayesian sampling plan.     

Calculation of Scores for a Wilcoxon Generalization Applicable to Data Subject to Arbitrary Right Censorship

For data subject to right censoring it is suggested that the Wilcoxon ranking procedure can be generalized by scoring observations according to the expected values of order statistics from the uniform distribution subject to the same right censoring. This parallels the logrank scoring procedure in which scores correspond to the expected values of order statistics from the exponential distribution that have been subject to right censoring. A caveat is given that, in situations where the mechanism of censoring has been affected by treatment, the usual permutational analysis of ranking scores would be inappropriate. But a jackknife approach could be remedial.     

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.     

Estimation of the smallest scale parameter of two-parameter exponential distributions

Improved point and interval estimation of the smallest scale parameter of n independent populations following two-parameter exponential distributions are studied. The model is formulated in such a way that allows for treating the estimation of the smallest scale parameter as a problem of estimating an unrestricted scale parameter in the presence of a nuisance parameter. The classes of improved point and interval estimators are enriched with Stein-type, Brewster and Zidek-type, Maruyama-type and Strawderman-type improved estimators under both quadratic and entropy losses, whereas using as a criterion the coverage probability, with Stein-type, Brewster and Zidek-type, and Maruyama-type improved intervals. The sampling framework considered incorporates important life-testing schemes such as i.i.d. sampling, type-II censoring, progressive type-II censoring, adaptive progressive type-II censoring, and record values.     

Bayesian estimation and prediction for the inverse weibull distribution under general progressive censoring

Abstract

This paper deals with Bayesian estimation and prediction for the inverse Weibull distribution with shape parameter α and scale parameter λ under general progressive censoring. We prove that the posterior conditional density functions of α and λ are both log-concave based on the assumption that λ has a gamma prior distribution and α follows a prior distribution with log-concave density. Then, we present the Gibbs sampling strategy to estimate under squared-error loss any function of the unknown parameter vector (α, λ) and find credible intervals, as well as to obtain prediction intervals for future order statistics. Monte Carlo simulations are given to compare the performance of Bayesian estimators derived via Gibbs sampling with the corresponding maximum likelihood estimators, and a real data analysis is discussed in order to illustrate the proposed procedure. Finally, we extend the developed methodology to other two-parameter distributions, including the Weibull, Burr type XII, and flexible Weibull distributions, and also to general progressive hybrid censoring.     

Statistical inference for the extreme value distribution under adaptive Type-II progressive censoring schemes

Adaptive Type-II progressive censoring schemes have been shown to be useful in striking a balance between statistical estimation efficiency and the time spent on a life-testing experiment. In this article, some general statistical properties of an adaptive Type-II progressive censoring scheme are first investigated. A bias correction procedure is proposed to reduce the bias of the maximum likelihood estimators (MLEs). We then focus on the extreme value distributed lifetimes and derive the Fisher information matrix for the MLEs based on these properties. Four different approaches are proposed to construct confidence intervals for the parameters of the extreme value distribution. Performance of these methods is compared through an extensive Monte Carlo simulation.     

Comparisons of approximate confidence intervals for the smallest extreme value distribution simple linear regression model under time censoring

Exact confidence interval estimation for accelerated life regression models with censored smallest extreme value (or Weibull) data is often impractical. This paper evaluates the accuracy of approximate confidence intervals based on the asymptotic normality of the maximum likelihood estimator, the asymptotic X²distribution of the likelihood ratio statistic, mean and variance correction to the likelihood ratio statistic, and the so-called Bartlett correction to the likelihood ratio statistic. The Monte Carlo evaluations under various degrees of time censoring show that uncorrected likelihood ratio intervals are very accurate in situations with heavy censoring. The benefits of mean and variance correction to the likelihood ratio statistic are only realized with light or no censoring. Bartlett correction tends to result in conservative intervals. Intervals based on the asymptotic normality of maximum likelihood estimators are anticonservative and should be used with much caution.     

A New Exponential GOF Test for Data Subject to Multiply Type II Censoring

This article presents a new goodness-of-fit (GOF) test statistic for multiply Type II censored Exponential data. The new test also applies to ordinary Type II censored samples and complete samples, since those cases are special cases of multiply Type II censoring. This test statistic is based on a ratio of linear functions of order statistics. Empirical power studies confirm that this ratio test compares favorably to currently available GOF tests for ordinary Type II censored data. Three data analysis examples are provided that demonstrate the usefulness of this new test statistic.     

Order statistics from overlapping samples: bivariate densities and regression properties

In this paper, we are interested in the joint distribution of two order statistics from overlapping samples. We give an explicit formula for the distribution of such a pair of random variables under the assumption that the parent distribution is absolutely continuous. We are also interested in the question to what extent conditional expectation of one of such order statistic given another determines the parent distribution. In particular, we provide a new characterization by linearity of regression of an order statistic from the extended sample given the one from the original sample, special case of which solves a problem explicitly stated in the literature. It appears that to describe the correct parent distribution it is convenient to use quantile density functions. In several other cases of regressions of order statistics we provide new results regarding uniqueness of the distribution in the sample.     

A new lifetime distribution for a series-parallel system: properties,applications and estimations under progressive type-II censoring

A compound class of zero truncated Poisson and lifetime distributions is introduced. A specialization is paved to a new three-parameter distribution, called doubly Poisson-exponential distribution, which may represent the lifetime of units connected in a series-parallel system. The new distribution can be obtained by compounding two zero truncated Poisson distributions with an exponential distribution. Among its motivations is that its hazard rate function can take different shapes such as decreasing, increasing and upside-down bathtub depending on the values of its parameters. Several properties of the new distribution are discussed. Based on progressive type-II censoring, six estimation methods [maximum likelihood, moments, least squares, weighted least squares and Bayes (under linear-exponential and general entropy loss functions) estimations] are used to estimate the involved parameters. The performance of these methods is investigated through a simulation study. The Bayes estimates are obtained using Markov chain Monte Carlo algorithm. In addition, confidence intervals, symmetric credible intervals and highest posterior density credible intervals of the parameters are obtained. Finally, an application to a real data set is used to compare the new distribution with other five distributions.     

Spacings,exceedances and concomitants in progressive type II censoring scheme

Distributional properties of spacings and exceedances based on progressive type II censored order statistics are investigated. Recurrence relations between moments of concomitants for the model of progressive type II censoring are given. We unify and extend some results on spacings, exceedances and concomitants for the model of progressive censoring.     

Inference for mixed generalized exponential distribution under progressively type-II censored samples

In industrial life tests, reliability analysis and clinical trials, the type-II progressive censoring methodology, which allows for random removals of the remaining survival units at each failure time, has become quite popular for analyzing lifetime data. Parameter estimation under progressively type-II censored samples for many common lifetime distributions has been investigated extensively. However, how to estimate unknown parameters of the mixed distribution models under progressive type-II censoring schemes is still a challenging and interesting problem. Based on progressively type-II censored samples, this paper addresses the estimation problem of mixed generalized exponential distributions. In addition, it is observed that the maximum-likelihood estimates (MLEs) cannot be easily obtained in closed form due to the complexity of the likelihood function. Thus, we make good use of the expectation-maximization algorithm to obtain the MLEs. Finally, some simulations are implemented in order to show the performance of the proposed method under finite samples and a case analysis is illustrated.