Inference based on progressively censored sample from Pareto population

Abstract

In this paper, we assume that the lifetimes have a two-parameter Pareto distribution and discuss some results of progressive Type-II censored sample. We obtain maximum likelihood estimators and Bayes estimators of the unknown parameters under squared error loss and a precautionary loss functions in progressively Type-II censored sample. Robust Bayes estimation of unknown parameters over three different classes of priors under progressively Type-II censored sample, squared error loss, and precautionary loss functions are obtained. We discuss estimation of unknown parameters on competing risks progressive Type-II censoring. Finally, we consider the problem of estimating the common scale parameter of two Pareto distributions when samples are progressively Type-II censored.     

Recurrence relations for moments of progressive type-II right censored order statstics from burr distribution

In this paper some recurrence relations between moments of progressive Type-II right censored order statistics from doubly truncated Burr distribution are established. These recurrence relations would enable one to obtain all the single and product moments of Burr progressive Type-II right censored order statistics in a simple recursive manner.     

Estimation for the Parameters of the Pareto Distribution Under Progressive Censoring

In this article, progressive Type-II right censored sample from Pareto distribution is considered. Exact confidence region is derived for the parameters of the corresponding distribution under progressive censoring. Simulation study is performed to investigate the coverage probabilities of the proposed confidence region. Illustrative example is also given.     

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.     

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.     

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.     

Exact inferences of the two-parameter exponential distribution and Pareto distribution with censored data

We develop an exact inference for the location and the scale parameters of the two-exponential distribution and the Pareto distribution based on their maximum-likelihood estimators from the doubly Type-II and the progressive Type-II censored sample. Based on some pivotal quantities, exact confidence intervals and tests of hypotheses are constructed. Exact distributions of the pivotal quantities are expressed as mixtures of linear combinations and of ratios of linear combinations of standard exponential random variables, which facilitates the computation of quantiles of these pivotal quantities. We also provide a bootstrap method for constructing a confidence interval. Some simulation studies are carried out to assess their performances. Using the exact distribution of the scale parameter, we establish an acceptance sampling procedure based on the lifetime of the unit. Some numerical results are tabulated for the illustration. One biometrical example is also given to illustrate the proposed methods.     

Recurrence relations for single and product moments of progressively Type-II censored order statistics from generalized logistic distribution with applications to inference

In this article, we establish several recurrence relations for the single and product moments of progressively Type-II right censored order statistics from a generalized logistic distribution. The use of these relations in a systematic manner allow us to compute all the means, variances, and covariances of progressively Type-II right censored order statistics from the generalized logistic distribution for all sample sizes n, effective sample sizes m, and all progressive censoring schemes (R₁, …, R_m). These moments are then utilized to derive best linear unbiased estimators of the scale and location-scale parameters of the generalized logistic distribution. A comparison of these estimators with the maximum likelihood estimates is then made through Monte Carlo simulations. Finally, the best linear unbiased predictors of censored failure times is discussed briefly.     

Recurrence relations for moments of progressively censored order statistics from logistic distribution with applications to inference

In this paper, we establish several recurrence relations for the single and product moments of progressively Type-II right censored order statistics from a logistic distribution. The use of these relations in a systematic manner allows us to compute all the means, variances and covariances of progressively Type-II right censored order statistics from the logistic distribution for all sample sizes n, effective sample sizes m, and all progressive censoring schemes (R₁,…,R_m). The results established here generalize the corresponding results for the usual order statistics due to [Shah, 1966] and [Shah, 1970]. These moments are then utilized to derive best linear unbiased estimators of the location and scale parameters of the logistic distribution. A comparison of these estimators with the maximum likelihood estimations is then made. The best linear unbiased predictors of censored failure times are briefly discussed. Finally, an illustrative example is presented.     

Nonparametric prediction of future order statistics

Prediction of censored order statistics from a Type-II censored sample can be done with trivial bounds having perfect confidence. However, given independent samples from the same absolutely continuous distribution, improved bounds can be attained. In this regard, we develop here point prediction based on L-statistics for predicting order statistics (OS) from a future sample as well as for predicting censored OS from a Type-II censored sample. An example is taken to illustrate these ideas, and the limiting case wherein a single independent sample is arbitrarily large is also discussed.     

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.     

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.     

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.     

On progressive hybrid censored exponential distribution

In this paper, we introduce a new adaptive Type-I progressive hybrid censoring scheme, which has some advantages over the progressive hybrid censoring schemes already discussed in the literature. Based on an adaptive Type-I progressively hybrid censored sample, we derive the exact distribution of the maximum-likelihood estimator (MLE) of the mean lifetime of an exponential distribution as well as confidence intervals for the failure rate using exact distribution, asymptotic distribution, and three parametric bootstrap resampling methods. Furthermore, we provide computational formula for the expected number of failures and investigate the performance of the point and interval estimation for the failure rate in this case. An alternative simple form for the distribution of the MLE under adaptive Type-II progressive hybrid censoring scheme proposed by Ng et al. [Statistical analysis of exponential lifetimes under an adaptive Type-II progressive censoring scheme, Naval Res. Logist. 56 (2009), pp. 687–698] is obtained. Finally, from the exact distribution of the MLE, we establish the explicit expression for the Bayes risk of a sampling plan under adaptive Type-II progressive hybrid censoring scheme when a general loss function is used, and present some optimal Bayes solutions under four different progressive hybrid censoring schemes to illustrate the effectiveness of the proposed method.     

On the existence of transformations preserving the structure of order statistics in lower dimensions

In a Type-II right censored sample from the standard uniform distribution, several transformations of respective order statistics are examined, which transform the censored sample into a complete sample in a lower dimension. Such transformations have been considered by Lin et al. (2008), Michael and Schucany (1979) and O’Reilly and Stephens (1988) in the context of goodness-of-fit tests. It is shown that by dropping the assumption of an underlying uniform distribution, these transformed random variables can no longer be considered themselves as order statistics, in general. In the case of the transformation of Michael and Schucany, it is shown that the uniform distribution is the only one possessing this property.     

Asymptotic properties of numbers of near minimum observations under progressive Type-II censoring

In this paper, some results of Pakes and Steutel [1997. On the number of records near the maximum, Austral. J. Statist. 39, 179–193] on the properties of the numbers of near order statistic observations are extended to the case of progressively Type-II censored data. In this way, we introduce the notion of the numbers of near minimum observations under progressive censoring. We derive distributional and asymptotic results for them and discuss the notion of the “increasing” sample in the case of progressive censoring. Some applications for the spacings, associated with the order statistics from progressively Type-II censored samples, are also provided.     

Exact distribution of the MLEs of the parameters and of the quantiles of two-parameter exponential distribution under hybrid censoring

Epstein [Truncated life tests in the exponential case, Ann. Math. Statist. 25 (1954), pp. 555–564] introduced a hybrid censoring scheme (called Type-I hybrid censoring) and Chen and Bhattacharyya [Exact confidence bounds for an exponential parameter under hybrid censoring, Comm. Statist. Theory Methods 17 (1988), pp. 1857–1870] derived the exact distribution of the maximum-likelihood estimator (MLE) of the mean of a scaled exponential distribution based on a Type-I hybrid censored sample. Childs et al. [Exact likelihood inference based on Type-I and Type-II hybrid censored samples from the exponential distribution, Ann. Inst. Statist. Math. 55 (2003), pp. 319–330] provided an alternate simpler expression for this distribution, and also developed analogous results for another hybrid censoring scheme (called Type-II hybrid censoring). The purpose of this paper is to derive the exact bivariate distribution of the MLE of the parameter vector of a two-parameter exponential model based on hybrid censored samples. The marginal distributions are derived and exact confidence bounds for the parameters are obtained. The results are also used to derive the exact distribution of the MLE of the pth quantile, as well as the corresponding confidence bounds. These exact confidence intervals are then compared with parametric bootstrap confidence intervals in terms of coverage probabilities. Finally, we present some numerical examples to illustrate the methods of inference developed here.     

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 generalized order statistics

A representation of the Fisher information in generalized order statistics in terms of the hazard rate of the underlying distribution function is derived under mild regularity conditions. This expression supplements results for complete, Type-II censored, and progressively Type-II censored data. As a byproduct, we find a hazard rate based representation for samples of k-records which apparently has not been known so far. Moreover, sufficient conditions for the validity of this representation in location and scale family settings are given. The result is illustrated by considering generalized order statistics based on logistic, Laplace, and extreme value distributions.     

Erratum to “Progressively Type-II right censored order statistics from discrete distributions” [J. Statist. Plann. Inference 138 (2008) 845–856]

In this note, we correct the proof of Representation 1 of Balakrishnan and Dembińska [2008. Progressively Type-II right censored order statistics from discrete distributions. J. Statist. Plann. Inference 138, 845–856] which relates the joint distribution of progressively Type-II right censored order statistics corresponding to an arbitrary population to progressively Type-II right censored order statistics from the standard uniform distribution.