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

In this paper, we consider two independent \(k\) -record sequences with the same distribution. We determine the closeness probability of \(k\) -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 future record intervals based on order statistics

In this article, outer and inner prediction intervals for future record intervals as well as record spacings are derived based on observed order statistics from the same parent distribution. These intervals are exact and are distribution-free in that they do not depend on the sampling distribution. Three different cases are considered and in each case an exact explicit expression is obtained for the prediction coefficient. Finally, we compare the obtained results with similar intervals based on records, and also present a numerical example in order to illustrate the derived results.     

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 X₁,X₂,…,X_n which are independent but not necessarily identically distributed (INID), and let K₁,K₂ be two integer-valued random variables, independent of {X₁,…,X_n}, such that 1≤K₁≤K₂≤n. It is shown that if K₁ has a log-concave probability function and SI(K₂|K₁) then RTI(X_K2:n|X_K1:n), and if K₂ has a log-concave probability function and SI(K₁|K₂) then LTD(X_K1:n|X_K2:n), where SI, RTI and LTD are three notions of bivariate positive dependence. Based on these, we obtain that RTI and LTD whenever 1≤i<j≤m, where are progressive Type-II censored order statistics from INID random variables {X₁,…,X_n}. Furthermore, one result concerning the likelihood ratio ordering of the progressive Type-II censored order statistics is also given.     

Two sample Bayesian prediction intervals for order statistics based on the inverse exponential-type distributions using right censored sample

In this paper, two sample Bayesian prediction intervals for order statistics (OS) are obtained. This prediction is based on a certain class of the inverse exponential-type distributions using a right censored sample. A general class of prior density functions is used and the predictive cumulative function is obtained in the two samples case. The class of the inverse exponential-type distributions includes several important distributions such the inverse Weibull distribution, the inverse Burr distribution, the loglogistic distribution, the inverse Pareto distribution and the inverse paralogistic distribution. Special cases of the inverse Weibull model such as the inverse exponential model and the inverse Rayleigh model are considered.     

Computational aspects of statistical intervals based on two Type-II censored samples

In this paper, we propose an efficient branch and bound procedure to compute exact nonparametric statistical intervals based on two Type-II right censored data sets. The procedure is based on some recurrence relations for the distribution and density functions of progressively Type-II censored order statistics which can be applied to compute the coverage probabilities. We illustrate the method for both confidence and prediction intervals of a given level.     

Recurrence relations for single and product moments of progressive Type-II right censored order statistics from exponential and truncated exponential distributions

In this paper, we establish several recurrence relations satisfied by the single and product moments of progressive Type-II right censored order statistics from an exponential distribution. These relations may then be used, for example, to compute all the means, variances and covariances of exponential progressive Type-II right censored order statistics for all sample sizes n and all censoring schemes (R ₁, R ₂, ..., R _m ), mn. The results presented in the paper generalize the results given by Joshi (1978, Sankhy Ser. B, 39, 362–371; 1982, J. Statist. Plann. Inference, 6, 13–16) for the single moments and product moments of order statistics from the exponential distribution.To further generalize these results, we consider also the right truncated exponential distribution. Recurrence relations for the single and product moments are established for progressive Type-II right censored order statistics from the right truncated exponential distribution.     

Confidence intervals based on interpolated order statistics

Confidence intervals for the population median based on interpolating adjacent order statistics are presented. They are shown to depend only slightly on the underlying distribution. A simple, nonlinear interpolation formula is given which works well for a broad collection of underlying distributions.     

Progressive Type II censored order statistics for multivariate observations

For a sequence of independent and identically distributed random vectors , i=1,2,…,n, we consider the conditional ordering of these random vectors with respect to the magnitudes of , where N is a p-variate continuous function defined on the support set of X₁ and satisfying certain regularity conditions. We also consider the Progressive Type II right censoring for multivariate observations using conditional ordering. The need for the conditional ordering of random vectors exists for example, in reliability analysis when a system has n independent components each consisting of p arbitrarily dependent and parallel connected elements. Let the vector of life lengths for the ith component of the system be , where denotes the life length of the jth element of the ith component. Then the first failure in the system occurs at time , and for this case . In this paper we introduce the conditionally ordered and Progressive Type II right-censored conditionally ordered statistics for multivariate observations and to study their distributional properties.     

Homogeneity tests based on several progressively Type-II censored samples

In this paper, we discuss the problem of testing the homogeneity of several populations when the available data are progressively Type-II censored. Defining for each sample a univariate counting process, we can modify all the methods that were developed during the last two decades (see e.g. [P.K. Andersen, Ø. Borgan, R. Gill, N. Keiding, Statistical Models Based on Counting Processes, Springer, New York, 1993]) for use to this problem. An important aspect of these tests is that they are based on either linear or non-linear functionals of a discrepancy process (DP) based on the comparison of the cumulative hazard rate (chr) estimated from each sample with the chr estimated from the whole sample (viz., the aggregation of all the samples), leading to either linear tests or non-linear tests. Both these kinds of tests suffer from some serious drawbacks. For example, it is difficult to extend non-linear tests to the K-sample situation when K?3. For this reason, we propose here a new class of non-linear tests, based on a chi-square type functional of the DP, that can be applied to the K-sample problem for any K?2.     

Prediction intervals based on Gompertz doubly censored data

This article is concerned with the problem of deriving Bayesian prediction bounds for the Gompertz \([Gomp(\alpha ,\,\beta )]\) distribution. Based on doubly Type II censored data, Bayesian prediction bounds for both the future observations will be derived. Two different sampling schemes have been considered. A conjugate prior for the one parameter case, as well as a joint prior for the two parameters case are outlined. Some numerical examples are given to illustrate the results.     

Confidence intervals based on steady-state continuous-time statistics

In steady-state computer simulation experiments, continuous-time statistics are used to estimate a variety of systems' parameters. In this paper we present formulas for estimating confidence intervals based on such statistics. Our formulas are extensions of the work of Schruben on ‘Standardized Time Series’ for discrete-time observations. We conduct a Monte Carlo study to examine the behavior of our estimators. Our estimators work quite well and our results are consistent with those of Schruben for discrete-time observations.     

Estimations from the exponentiated rayleigh distribution based on generalized Type-II hybrid censored data

In this article, unknown parameters of exponentiated Rayleigh distribution based on generalized Type II Hybrid censored data, survival function, failure rate function and coefficient of variation are derived by applying the maximum likelihood, Bayes and percentile bootstrap methods. Approximate confidence intervals for the unknown parameters, survival function, failure rate function and coefficient of variation are obtained. We study Bayes estimates under gamma priors distributions depending on symmetric and asymmetric loss functions via the Gibbs within Metropolis-Hasting samplers procedure. Finally, the proposed methods can be understood through illustrating the results of the real data analysis.     

Nonparametric estimation based on censored observations of a Markov renewal process

Summary Uniform consistency and weak convergence is proved of estimators of the transition probabilities of an arbitrary finite state space Markov renewal process, based on n independent and identically distributed right censored realizations of the process. The approach uses the theory of stochastic integrals and counting processes. It is shown how the results may be extended to the non-identically distributed case and to general censorship under suitable conditions.     

Conservative confidence intervals based on weighted means statistics

For weighted means estimators of the common mean of several normal populations associated (conservative) confidence intervals are constructed. These intervals are compared to several traditional confidence bounds. Monte Carlo simulation results of these comparisons are reported.     

Asymptotic linear prediction of extreme order statistics

Summary We consider the problem of predicting thesth order statistic using the lowestr order statistics from a large sample of sizen under the assumption that the sample minimum, appropriately normalized, has a non-degenerate limit distribution asn→∞. Assumingr, s fixed andn→∞ we obtain asymptotically best linear unbiased as well as asymptotically best linear invariant predictors of thesth order statistic.     

Mixture representation for order statistics from INID progressive censoring and its applications

In this paper, a mixture representation for the joint distribution function of progressively Type-II censored order statistics from heterogeneous distributions is established. Applications of this representation to stochastic orderings and inequalities are then illustrated.     

An efficient computational method for moments of order statistics under progressive censoring

Thomas and Wilson (Technometrics 14 (1972) 679) developed a computational method for calculating the single and product moments of order statistics from progressively censored samples by making use of the corresponding moments of the usual order statistics. The absence of an explicit representation for the marginal and joint density function of order statistics under progressive censoring makes their method extremely tedious. By deriving the required marginal and joint density functions in explicit form, we obtain an alternative, highly efficient, method for computing the desired moments.     

Exact likelihood inference based on Type-I and Type-II hybrid censored samples from the exponential distribution

Chen and Bhattacharyya (1988,Comm. Statist. Theory Methods,17, 1857–1870) derived the exact distribution of the maximum likelihood estimator of the mean of an exponential distribution and an exact lower confidence bound for the mean based on a hybrid censored sample. In this paper, an alternative simple form for the distribution is obtained and is shown to be equivalent to that of Chen and Bhattacharyya (1988). Noting that this scheme, which would guarantee the experiment to terminate by a fixed timeT, may result in few failures, we propose a new hybrid censoring scheme which guarantees at least a fixed number of failures in a life testing experiment. The exact distribution of the MLE as well as an exact lower confidence bound for the mean is also obtained for this case. Finally, three examples are presented to illustrate all the results developed here.