Progressive Type-II Censored Order Statistics Research Articles

Measures of Extropy Based on Concomitants of Generalized Order Statistics under a General Framework from Iterated Morgenstern Family

In this work, we reveal some distributional characteristics of concomitants of generalized order statistics (GOS) with parameters that are pairwise different, arising from iterated Farlie–Gumbel–Morgenstern (IFGM) family of bivariate distributions. Additionally, the joint distribution and product moments of concomitants of GOS for this family are discussed. Moreover, some well-known information measures, i.e., extropy, cumulative residual extropy (CRJ), and negative cumulative extropy (NCJ), are derived. Applications of these results are given for order statistics, record values, and progressive type-II censored order statistics with uniform marginals distributions. Additionally, the issue of estimating the CRJ and NCJ is looked into, utilizing the empirical technique and the concomitant of GOS. Finally, bivariate real-world data sets have been analyzed for illustrative purposes, and the performance of the proposed method is quite satisfactory.

Cumulative entropy of progressively type-II censored order statistics and associated optimal life testing-plans

In this paper, joint cumulative entropy of progressive type-II censored order statistics is obtained in terms of reversed hazard rate function. Also, entropy and Kullback–Leibler information for progressive type-II censored order statistics are expressed using reversed hazard rate. Optimal life-testing plans based on cumulative entropy are proposed. Finally, cumulative entropy-based optimal designs using compound optimal design strategy for the life-testing experiment under progressive type-II censoring are developed. A real data set is analysed for illustration.

On concomitants of generalized order statistics from generalized FGM family under a general setting

AbstractIn this work, we study the distributional and moment properties of concomitants of dual generalized order statistics and consequently generalized order statistics. When the parameters are assumed to be pairwise different from Bairamov-Kotz-Becki Farlie-Gumbel-Morgenstern (BKB-FGM) family. Furthermore, an example of progressive type-II censored order statistics of BKB-FGM family under uniform distribution is obtained. Moreover, the joint distribution of dual generalized order statistics and record values of concomitants for this family are discussed. Finally, an application is given for some well-known distributions such as exponential, Pareto, and power function distributions.

Likelihood ratio comparisons and logconvexity properties ofp-spacings from generalized order statistics

Due to the importance of generalized order statistics (GOS) in many branches of Statistics, a wide interest has been shown in investigating stochastic comparisons of GOS. In this article, we study the likelihood ratio ordering of$p$-spacings of GOS, establishing some flexible and applicable results. We also settle certain unresolved related problems by providing some useful lemmas. Since we do not impose restrictions on the model parameters (as previous studies did), our findings yield new results for comparison of various useful models of ordered random variables including order statistics, sequential order statistics,$k$-record values, Pfeifer's record values, and progressive Type-II censored order statistics with arbitrary censoring plans. Some results on preservation of logconvexity properties among spacings are provided as well.

Information Measures for Generalized Order Statistics and Their Concomitants under General Framework from Huang-Kotz FGM Bivariate Distribution.

In this paper, we study the concomitants of dual generalized order statistics (and consequently generalized order statistics) when the parameters are assumed to be pairwise different from Huang–Kotz Farlie–Gumble–Morgenstern bivariate distribution. Some useful recurrence relations between single and product moments of concomitants are obtained. Moreover, Shannon’s entropy and the Fisher information number measures are derived. Finally, these measures are extensively studied for some well-known distributions such as exponential, Pareto and power distributions. The main motivation of the study of the concomitants of generalized order statistics (as an important practical kind to order the bivariate data) under this general framework is to enable researchers in different fields of statistics to use some of the important models contained in these generalized order statistics only under this general framework. These extended models are frequently used in the reliability theory, such as the progressive type-II censored order statistics.

On properties of progressively Type-II censored conditionally N-ordered statistics arising from a non-identical and dependent random vector

In some systems, we observe the lifetimes of its components but the classical model based on progressively Type-II censored order statistics (PCOS-II) do not use this information to analysis the lifetime of the system. For example, when we use PCOS-II to consider the lifetime of a parallel system we do not use the lifetime of each component. In fact, we just use the lifetime of the system which equals the maximum lifetimes of its component. Some other example of these systems in which we do not use all the information that we observed during the lifetime testing procedure based on PCOS-II are repairable system and a system with n stand-by components. Instead of it, if we use progressively Type-II censored conditionally N-ordered statistics (PCCOS-N) to analysis the lifetime of these systems we can use all the information that we observed and obtained a more accurate analysis for lifetime of the system. PCCOS-N is introduced by Bairamov.[Progressive Type II censored order statistics for multivariate observations. J Mult Anal. 2006;97:797–809] In this paper, we investigate PCCOS-N arising from a system with identical as well as non-identical but dependent components, jointly distributed according to an Archimedean copula with completely monotone generator (PCCOSDNARCM-N). Our results generalized the results in Bairamov (2006) and is more flexible than those in practice, because of considering the dependency between components that is a common fact for real data. A simulation study with a method of parameter estimation is also finally provided.

Closeness of k-records to Progressive Type-II Censored Order Statistics for Location-scale Families

In this article, the Pitman closeness of upper and lower k-records to progressive Type-II censored order statistics for location-scale families is investigated. In each case, the special properties of the probability of Pitman closeness are obtained and the corresponding monotonicity properties are discussed. Moreover, the closest k-record to a specific progressive Type-II censored data is obtained. Finally, for the standard exponential and standard uniform distributions, explicit expressions for the probability of Pitman closeness are derived. For various censoring schemes, the results of the numerical computations are displayed in tables. Most of the results in Ahmadi and Balakrishnan (2013) can be achieved as special cases.

Pitman closeness of $$k$$ k -records from two sequences to progressive Type-II censored order statistics

In this paper, we consider two independent $$k$$ k -record sequences with the same distribution. We determine the closeness probability of $$k$$ k -record values to a specific progressive Type-II censored order statistic. With this in mind, we first derive the exact expression for the Pitman closeness of records in general, and some special properties of the closeness probability are presented. Then, we apply the obtained results for the standard uniform and exponential distributions and exact expressions for the Pitman closeness are obtained. Finally, numerical results are displayed in figures.

Nonparametric prediction intervals for progressive Type-II censored order statistics based on $$k$$ k -records

In this paper, we consider the prediction problem in two-sample case and study the non-parametric predicting future progressively Type-II censored order statistics based on observed $$k$$ k -records from the same distribution. Also, prediction intervals for progressively Type-II censored spacings are obtained based on $$k$$ k -record spacings. It is shown that the coverage probabilities of these intervals are exact and do not depend on the underlying distribution. Moreover, optimal prediction intervals are derived for each case. Finally, for illustrating the proposed procedure, we consider a real data set and numerical computations are given. The results of Ahmadi and Balakrishnan (Statistics 44:417---430, 2010) can be achieved as special cases of our results.

Moments from Progressive Type-II Censored Data of Marshall-Olkin Exponential Distribution

Marshall-Olkin exponential distribution has been studied by Salah, et al. [6]. In this paper, we derive several recurrence relations satised by the single moments and product moments of progressive type-II censored order statistics from Marshall-Olkin exponential distribution. These relations may then be used to compute all means, variances and covariances of progressive type-II censored ordered statistics based on Marshall-Olkin exponential distribution for arbitrary censoring scheme.

Estimation Based on Generalized Order Statistics From a Mixture of Two Rayleigh Distributions

This article is concerned with the problem of estimating the parameters, reliability and hazard rate functions of the mixture of two Rayleigh distributions ($MTRD$) based on generalized order statistics ($GOS$). The maximum likelihood and Bayes methods of estimation are used for this purpose. The Markov chain Monte Carlo ($MCMC$) method is used for obtaining the Bayes estimates under the squared error loss and $LINEX$ loss functions. Our results are specialized to progressive Type-II censored order statistics and upper record values. Comparisons are made between Bayesian and maximum likelihood estimators via a Monte Carlo simulation study.

Stochastic properties of INID progressive Type-II censored order statistics

Let X 1 : n ≤ X 2 : n ≤ ⋯ ≤ X n : n denote the order statistics of random variables X 1 , X 2 , … , X n which are independent but not necessarily identically distributed (INID), and let K 1 , K 2 be two integer-valued random variables, independent of { X 1 , … , X n } , such that 1 ≤ K 1 ≤ K 2 ≤ n . It is shown that if K 1 has a log-concave probability function and SI ( K 2 | K 1 ) then RTI ( X K 2 : n | X K 1 : n ) , and if K 2 has a log-concave probability function and SI ( K 1 | K 2 ) then LTD ( X K 1 : n | X K 2 : n ) , where SI, RTI and LTD are three notions of bivariate positive dependence. Based on these, we obtain that RTI ( X j : m , n R | X i : m , n R ) and LTD ( X i : m , n R | X j : m , n R ) whenever 1 ≤ i < j ≤ m , where { X 1 : m , n R , … , X m : m , n R } are progressive Type-II censored order statistics from INID random variables { X 1 , … , X n } . Furthermore, one result concerning the likelihood ratio ordering of the progressive Type-II censored order statistics is also given.

Comments on: Progressive censoring methodology: an appraisal

Professor Balakrishnan provides a very clear and elaborate review of various theoretical results concerning progressively Type-II censored order statistics and their applications to inferential problems. An extensive list of references assists with further reading. Moreover, diverse open problems included in the text stimulate the reader to join the research on progressive censoring. My attention has been drawn specifically to Open Problem 2. In the literature different types of bounds for moments of usual order statistics and their functions are known. A class of such bounds was obtained by an application of the CauchySchwarz inequality. In the case, when this approach does not lead to an attainable bound, an improvement can be achieved through an elegant method of “greatest convex minorant” due to Moriguti (1953), see Balakrishnan (1993). As discussed briefly in Sect. 7.2 of Professor Balakrishnan’s review, Balakrishnan et al. (2001) generalized these bounds to the case of progressively Type-II right censored order statistics. In the same manner, but using the Holder inequality, Raqab (2003) derived upper bounds for moments of progressive Type-II censored order statistics in various scale units generated by central absolute moments. It should be mentioned here that Raqab’s (2003) bounds are special cases of more general results established earlier by Cramer et al. (2002) in the context of generalized order statistics. In his review Professor Balakrishnan poses a question, whether other bounds for moments of usual order statistics or their functions could be generalized to the case of progressive Type-II censoring situation. Yet, I have noticed that even the already known generalizations by Balakrishnan et al. (2001) and Raqab (2003) are not complete in the sense that bounds for

A CHARACTERIZATION OF GENERALIZED PARETO DISTRIBUTIONS BY PROGRESSIVE CENSORING SCHEMES AND GOODNESS-OF-FIT TESTS

ABSTRACT In this paper we generalize a characterization property of generalized Pareto distributions, which is known for ordinary order statistics, to arbitrary schemes of progressive Type-II censored order statistics. Various goodness-of-fit tests for generalized Pareto distributions based on progressively censored data statistics are discussed.

Some properties of progressive censored order statistics from arbitrary and uniform distributions with applications to inference and simulation

In this paper, we first establish three properties of progressive Type-II censored order statistics from arbitrary continuous distributions. These properties are then used to develop an algorithm to simulate general progressive Type-II censored order statistics from any continuous distribution, by generalizing the algorithm given recently by Balakrishnan and Sandhu (Sankhya Series B58 (1995), 1–9). We then establish an independence result for general progressive Type-II censored samples from the uniform (0,1) population, which generalizes a result given by Balakrishnan and Sandhu (1995) for progressive Type-II right censored samples. This result is used in order to obtain moments for general progressive Type-II censored order statistics from the uniform (0,1) distribution. This independence result also gives rise to a second algorithm for the generation of general progressive Type-II censored order statistics from any continuous distribution. Finally, best linear unbiased estimators (BLUEs) for the parameters of one- and two-parameter uniform distributions are derived, and the problem of maximum-likelihood estimation is discussed.