Recurrence relations for single and product moments of progressively Type-II censored order statistics from a generalized half-logistic distribution with application 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 generalized half-logistic distribution. The use of these relations in a systematic recursive manner enables the computation of all the means, variances, and covariances of progressively Type-II right-censored order statistics from the generalized half-logistic distribution for all sample sizes n, effective sample sizes m, and all progressive censoring schemes (R ₁, …, R _m ). The results established here generalize the corresponding results for the usual order statistics due to Balakrishnan and Sandhu [Recurrence relations for single and product moments of order statistics from a generalized half-logistic distribution with applications to inference, J. Stat. Comput. Simul. 52 (1995), pp. 385–398.]. The moments so determined are then utilized to derive the best linear unbiased estimators of the scale and location–scale parameters of the generalized half-logistic distribution. The best linear unbiased predictors of censored failure times are discussed briefly. Finally, a numerical example is presented to illustrate the inferential method developed here.     

Extreme Value Analysis for Progressively Type-II Censored Order Statistics

An account to extreme value theory for progressively Type-II censored order statistics is presented which enables us to handle limit laws for upper and lower extreme, intermediate and central progressively Type-II censored order statistics within one framework. We illustrate that the extreme value analysis for progressively Type-II censored order statistics is connected to limit laws for sums of independent but not-identically distributed exponential random variables. Moreover, we show that the limits are transformations of extreme value distributions and illustrate the connection to extreme value analysis for order statistics.     

Simultaneous Pitman closeness of progressively type-II right-censored order statistics to population quantiles

In this work, we extend prior results concerning the simultaneous Pitman closeness of order statistics (OS) to population quantiles. By considering progressively type-II right-censored samples, we derive expressions for the simultaneous closeness probabilities of the progressively censored OS to population quantiles. Explicit expressions are deduced for the cases when the underlying distribution has bounded and unbounded supports. Illustrations are provided for the cases of exponential, uniform and normal distributions for various progressive type-II right-censoring schemes and different quantiles. Finally, an extension to the case of generalized OS is outlined.     

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.     

On Distributions of Exceedances Based on Generalized Order Statistics

Distributions of exceedance statistics based on generalized order statistics are obtained for a random threshold model. The ordinary order statistics, progressively Type-II right censored order statistics and record values are considered as special cases. The results obtained in the article imply many results on exceedance statistics for the variety of models of ordered random variables.     

Recursive computation of the single and product moments of order statistics from the complementary exponential–geometric distribution

The complementary exponential–geometric distribution has been proposed recently as a simple and useful reliability model for analysing lifetime data. For this distribution, some recurrence relations are established for the single and product moments of order statistics. These recurrence relations enable the computation of the means, variances and covariances of all order statistics for all sample sizes in a simple and efficient recursive manner. By using these relations, we have tabulated the means, variances and covariances of order statistics from samples of sizes up to 10 for various values of the shape parameter θ. These values are in turn used to determine the best linear unbiased estimator of the scale parameter β based on complete and Type-II right-censored samples.     

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.     

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.     

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.     

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.     

Bayesian reconstruction of the missing failure times in exponential distribution

In this paper, we consider the situations in which a middle part of the sample is lost or removed from the experiment. The left-censored sample and Type-II right-censored sample are special cases of this scheme. Point and interval reconstructors for lost failure times are obtained, while the underlying distribution is exponential. To illustrate the performance of the proposed procedures, two numerical examples based on the real data sets are presented. In the case of two-parameter, a simulation study has been done and also the sensitivity of the results to deviations from the true distribution is investigated.     

Goodness-of-fit tests for progressively Type-II censored data from location–scale distributions

In this article, we propose several goodness-of-fit methods for location–scale families of distributions under progressively Type-II censored data. The new tests are based on order statistics and sample spacings. We assess the performance of the proposed tests for the normal and Gumbel models against several alternatives by means of Monte Carlo simulations. It has been observed that the proposed tests are quite powerful in comparison with an existing goodness-of-fit test proposed for progressively Type-II censored data by Balakrishnan et al. [Goodness-of-fit tests based on spacings for progressively Type-II censored data from a general location–scale distribution, IEEE Trans. Reliab. 53 (2004), pp. 349–356]. Finally, we illustrate the proposed goodness-of-fit tests using two real data from reliability literature.     

More on progressively Type‐II censored order statistics for multivariate observations

In this paper, we consider some results on distribution theory of multivariate progressively Type‐II censored order statistics. We also establish some characterizations of Freund's bivariate exponential distribution based on the lack of memory property.     

A NOTE ON MOMENTS OF PROGRESSIVELY TYPE II CENSORED ORDER STATISTICS

ABSTRACT

Upper and lower bounds for moments of progressively Type II censored order statistics in terms of moments of (progressively Type II censored) order statistics are derived. In particular, this yields conditions for the existence of moments of progressively Type II censored order statistics based on an absolutely continuous distribution function.     

On distributions Of generalized order statistics

In a wide subclass of generalized order statistics, representations of marginal density and distribution functions are developed. The results are applied to obtain several relations, such as recurrence relations, and explicit expressions for the moments of generalized order statistics from Pareto, power function and Weibull distributions Moreover, characterizations of exponential distributions are shown by means of a distributional identity as well as by* an identity of expectations involving a subrange and a corresponding generalized order statistic.     

Higher order moments of order statistics from the Lindley distribution and associated inference

ABSTRACT

The Lindley distribution is an important distribution for analysing the stress–strength reliability models and lifetime data. In many ways, the Lindley distribution is a better model than that based on the exponential distribution. Order statistics arise naturally in many of such applications. In this paper, we derive the exact explicit expressions for the single, double (product), triple and quadruple moments of order statistics from the Lindley distribution. Then, we use these moments to obtain the best linear unbiased estimates (BLUEs) of the location and scale parameters based on Type-II right-censored samples. Next, we use these results to determine the mean, variance, and coefficients of skewness and kurtosis of some certain linear functions of order statistics to develop Edgeworth approximate confidence intervals of the location and scale Lindley parameters. In addition, we carry out some numerical illustrations through Monte Carlo simulations to show the usefulness of the findings. Finally, we apply the findings of the paper to some real data set.     

Some distributional problems connected with order statistics

Two characterizations of distributions symmetric about zero are given. These are based on the distributional properties of the squates of the order statistics from a random sample from these distributions. A result explering the relation between the distribution funcitons of two unordered (not necessarily independent) variables and those of their order statistics is presented. This has some interesting applications.     

Progressively Type-II right censored order statistics from discrete distributions

Progressively Type-II right censored order statistics from continuous distributions have been studied rather extensively in the literature; see Balakrishnan and Aggarwala [2000. Progressive Censoring: Theory, Methods and Applications. Birkhäuser, Boston]. In this paper, we derive the joint and marginal distributions of progressively Type-II right censored order statistics from discrete distributions. We then use these distributions to show the non-Markovian property as well as to discuss some properties in the special case of the geometric distribution.     

Bayesian Estimation for the Exponentiated Weibull Model via Markov Chain Monte Carlo Simulation

Bayesian estimation for the two unknown parameters and the reliability function of the exponentiated Weibull model are obtained based on generalized order statistics. Markov chain Monte Carlo (MCMC) methods are considered to compute the Bayes estimates of the target parameters. Our computations are based on the balanced loss function which contains the symmetric and asymmetric loss functions as special cases. The results have been specialized to the progressively Type-II censored data and upper record values. Comparisons are made between Bayesian and maximum likelihood estimators via Monte Carlo simulation.     

Distribution of the Number of Observations Greater Than the ith Dependent Progressively Type-II Censored Order Statistic and Its Use in Goodness-of-fit Testing

In a life-testing problem, it may be of interest to investigate the number of observations close to but greater than the median, minimum, or more generally, the ith progressively Type-II censored order statistic (PCOS-II). In this paper, we derive the probability mass and distribution functions of the number of observations greater than the ith PCOS-II for a system with identical components for the cases of independent as well as dependent components. The type of dependence considered among the component lifetimes is through an Archimedean copula. We also provide a goodness-of-fit method for determining the best copula model for a given PCOS-II. Finally, an example is provided to illustrate the results developed here.