On generalized conditional cumulative residual inaccuracy measure

Abstract

Recently, the notion of cumulative residual Rényi’s entropy has been proposed in the literature as a measure of information that parallels Rényi’s entropy. Motivated by this, here we introduce a generalized measure of it, namely cumulative residual inaccuracy of order α. We study the proposed measure for conditionally specified models of two components having possibly different ages called generalized conditional cumulative residual inaccuracy measure. Several properties of generalized conditional cumulative residual inaccuracy measure including the effect of monotone transformation are investigated. Further, we provide some bounds on using the usual stochastic order and characterize some bivariate distributions using the concept of conditional proportional hazard rate model.     

Characterizations of discrete distributions based on conditional expectations of record values

r -th record values subject to (r + 1)-th record values, record mean function, from a distribution of discrete type. We give some properties of the record mean function and an explicit expression for the distribution function based on its record mean function, which allows us to characterize particular discrete distributions using the record mean functions. Received: January 4, 1999; revised version: September 27, 1999     

Comparisons of several Pareto distributions based on record values

In order to avoid wrong conclusions in any further analysis, it is of importance to conduct a formal comparison for characteristic quantities of the distributions. These characteristic quantities we are familiar with include mean, quantity and reliability function, and so on. In this paper, we consider two tests aiming at the comparisons for function of parameters in Pareto distribution based on record values. They are generalized p-value-based test and parametric bootstrap-based test, respectively. The resulting procedures are easy to compute and are applicable to small samples. A simulation study is conducted to investigate and compare the performance of the proposed tests. A phenomenon we note is that generalized p-value-based test almost uniformly outperforms the parametric bootstrap-based test.     

Shortest tolerance intervals controlling both tails of the exponential distribution based on record values

This paper deals with computing shortest width tolerance intervals controlling both tails of the exponential distribution on the basis of record values. Equal-tailed and shortest tolerance factors are derived. The expected widths of these tolerance intervals are evaluated via a Monte Carlo simulation study. Finally, two illustrative examples are also included.     

Estimation of parameters of bivariate normal distribution using concomitants of record values

In this paper, we discuss the concomitants of record values arising from the well-known bivariate normal distribution BVND(μ₁, μ₂,σ₁,σ₂, ρ). We have obtained the best linear unbiased estimators of μ₂ and σ₂ when ρ is known and derived some unbiased linear estimators of ρ when μ₂ and σ₂ are known, based on the concomitants of first n record values. The variances of these estimators have been obtained.     

On weighted measure of inaccuracy for doubly truncated random variables

Recently, authors have studied the weighted version of Kerridgeinaccuracy measure for left/right truncated distributions. In the present communication we introduce the notion of weighted interval inaccuracy measure and study it in the context of two-sided truncated random variables. In reliability theory and survival analysis, this measure may help to study the various characteristics of a system/component when it fails between two time points. We study various properties of this measure, including the effect of monotone transformations, and obtain its upper and lower bounds. It is shown that the proposed measure can uniquely determine the distribution function and characterizations of some important life distributions have been provided. This new measure is a generalization of recent weighted residual (past) inaccuracy measure.     

Concomitant record ranked set sampling

In this work, we define a new method of ranked set sampling (RSS) which is suitable when the characteristic (variable) Y of primary interest on the units is jointly distributed with an auxiliary characteristic X on which one can take its measurement on any number of units, so that units having record values on X alone are ranked and retained for making measurement on Y. We name this RSS as concomitant record ranked set sampling (CRRSS). We propose estimators of the parameters associated with the variable Y of primary interest based on observations of the proposed CRRSS which are applicable to a very large class of distributions viz. Morgenstern family of distributions. We illustrate the application of CRRSS and our estimation technique of parameters, when the basic distribution is Morgenstern-type bivariate logistic distribution. A primary data collected by CRRSS method is demonstrated and the obtained data used to illustrate the results developed in this work.     

Statistical inference based on generalized Lindley record values

This paper addresses the problems of frequentist and Bayesian estimation for the unknown parameters of generalized Lindley distribution based on lower record values. We first derive the exact explicit expressions for the single and product moments of lower record values, and then use these results to compute the means, variances and covariance between two lower record values. We next obtain the maximum likelihood estimators and associated asymptotic confidence intervals. Furthermore, we obtain Bayes estimators under the assumption of gamma priors on both the shape and the scale parameters of the generalized Lindley distribution, and associated the highest posterior density interval estimates. The Bayesian estimation is studied with respect to both symmetric (squared error) and asymmetric (linear-exponential (LINEX)) loss functions. Finally, we compute Bayesian predictive estimates and predictive interval estimates for the future record values. To illustrate the findings, one real data set is analyzed, and Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation and prediction.     

Reliability properties of record values from non-identically distributed random variables

In reliability theory, order statistics and record values are used for statistical modelling. The r-th order statistic in a sample of size n represents the life—length of a (n?r+l)-out-of-n system, and record values are used in shock models. In recent years, reliability properties of order statistics and record values have been investigated. The two models are included in Pfeifer's concept of record values from non-identically distributed random variables. Here, some results on the transmission of distributional properties, such as increasing failure rate, are shown for such records, which contain the results for order statistics and ordinary record values as particular cases.     

On k-th record times,record values and their moments

We give new formula for moments of k-th record values in terms of Stirling numbers of the first kind. In particular, the formulae allow to derive the explicit formulae for moments of k-th lower record values from exponential distribution which have not been known yet. Moreover, some interesting identities involving harmonic numbers are also obtained as corollaries to presented results.     

Asymptotic distributions of order statistics and record values arising from the class of beta-generated distributions

In this paper, we show that both the class of beta-generated distributions G_F and its base distribution F belong to the same domain of maximal (or minimal or upper record value or lower record value) attraction. Moreover, it is shown that the weak convergence of any non-extreme order statistic (central or intermediate order statistic), based on a base distribution F, to a non-degenerate limit type implies the weak convergence of G_F to a non-degenerate limit type. The relations between the two limit types are deduced.     

Prediction of record values

Given a number of record values from independent and identically distributed random variables with a continuous distribution function F, our aim is to predict future record values under suitable assumptions on the tail of F. In this paper, we are primarily concerned with finding reasonable tolerance regions for future record values. Two methods are proposed. The first one deals with the case where we observe only record values. The second one makes use of the information brought by the complete sample.     

On diagnostic devices for proposing half-logistic and inverse half-logistic models using generalized (k) record values

In this work we consider the generalized upper (k) record values (GURV’s) and generalized lower (k) record values (GLRV’s) arising from half-logistic distribution (HLD) and inverse half-logistic distribution (IHLD). We derive some characterization results of HLD based on some moment relations of generalized upper (k) record values and those of generalized lower (k) record values and accordingly devised some diagnostic tools to identify HLD as a model to the distribution of a population. Similar characterization theorems and diagnostic tools are developed for IHLD as well. Simulation studies are conducted to validate the diagnostic tools devised for both HLD and IHLD.     

Bayesian analysis for the two-parameter Pareto distribution based on record values and times

Doostparast and Balakrishnan (Pareto record-based analysis, Statistics, under review) recently developed optimal confidence intervals as well as uniformly most powerful tests for one- and two-sided hypotheses concerning shape and scale parameters, for the two-parameter Pareto distribution based on record data. In this paper, on the basis of record values and inter-record times from the two-parameter Pareto distribution, maximum-likelihood and Bayes estimators as well as credible regions are developed for the two parameters of the Pareto distribution. For illustrative purposes, a data set on annual wages of a sample of production-line workers in a large industrial firm is analysed using the proposed procedures.     

On distributions of exceedances associated with order statistics and record values for arbitrary distributions

Received: October 12, 1999; revised version: April 6, 2000     

Statistical inference about the shape parameter of the Weibull distribution by upper record values

Summary In this paper, we provide some pivotal quantities to test and establish confidence interval of the shape parameter on the basis of the firstn observed upper record values. Finally, we give some examples and the Monte Carlo simulation to assess the behaviors (including higher power and more shorter length of confidence interval) of these pivotal quantities for testing null hypotheses and establishing confidence interval concerning the shape parameter under the given significance level and the given confidence coefficient, respectively.     

Stress–strength reliability of a non-identical-component-strengths system based on upper record values from the family of Kumaraswamy generalized distributions

In an attempt to produce more realistic stress–strength models, this article considers the estimation of stress–strength reliability in a multi-component system with non-identical component strengths based on upper record values from the family of Kumaraswamy generalized distributions. The maximum likelihood estimator of the reliability, its asymptotic distribution and asymptotic confidence intervals are constructed. Bayes estimates under symmetric squared error loss function using conjugate prior distributions are computed and corresponding highest probability density credible intervals are also constructed. In Bayesian estimation, Lindley approximation and the Markov Chain Monte Carlo method are employed due to lack of explicit forms. For the first time using records, the uniformly minimum variance unbiased estimator and the closed form of Bayes estimator using conjugate and non-informative priors are derived for a common and known shape parameter of the stress and strength variates distributions. Comparisons of the performance of the estimators are carried out using Monte Carlo simulations, the mean squared error, bias and coverage probabilities. Finally, a demonstration is presented on how the proposed model may be utilized in materials science and engineering with the analysis of high-strength steel fatigue life data.     

Estimation with the generalized exponential distribution based on record values and inter-record times

The maximum likelihood and Bayesian approaches have been considered for the two-parameter generalized exponential distribution based on record values with the number of trials following the record values (inter-record times). The maximum likelihood estimates are obtained under the inverse sampling and the random sampling schemes. It is shown that the maximum likelihood estimator of the shape parameter converges in mean square to the true value when the scale parameter is known. The Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo methods due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The confidence intervals for the parameters are constructed based on asymptotic and Bayesian methods. The Bayes and the maximum likelihood estimators are compared in terms of the estimated risk by the Monte Carlo simulations. The comparison of the estimators based on the record values and the record values with their corresponding inter-record times are performed by using Monte Carlo simulations.     

Lower bounds on the expectations of upper record values

For infinite sequences of independent random variables with identical continuous distributions, we establish optimal lower bounds on the deviations of the expectations of record values from population means in units generated by the central absolute moments of various orders. The bounds are non-negative for the classic record values, and non-positive for the other kth records with k?2. We also provide analogous bounds for the record increments.     

Characterization of discrete distributions using expected values

In this paper we study characterization problems for discrete distributions using the doubly truncated mean function m(xy)=E(h(X)|x≤X≤y), for a monotonic function h(x). We obtain the distribution function F(x) from m(x,y) and we give the necessary and sufficient conditions for any real function to be the doubly truncated mean function for a discrete distribution.