A new measure of association for bivariate survival data

Time dependent association measures between variables are of interest in bivariate survival data. Several such measures have been proposed in literature for the modelling and analysis of survival data. In this paper, we introduce a new measure of association for bivariate survival data using product moment residual life function and mean residual life function. Various properties of the proposed measure and its relationship with existing measures are discussed. We also develop a non-parametric estimator of the measure and study its asymptotic properties. The application of the result is illustrated using a real life data. Finally, a stimulation study is carried out to assess the performance of the estimator.     

A general quantile residual life model for length‐biased right‐censored data

The quantile residual lifetime function provides comprehensive quantitative measures for residual life, especially when the distribution of the latter is skewed or heavy‐tailed and/or when the data contain outliers. In this paper, we propose a general class of semiparametric quantile residual life models for length‐biased right‐censored data. We use the inverse probability weighted method to correct the bias due to length‐biased sampling and informative censoring. Two estimating equations corresponding to the quantile regressions are constructed in two separate steps to obtain an efficient estimator. Consistency and asymptotic normality of the estimator are established. The main difficulty in implementing our proposed method is that the estimating equations associated with the quantiles are nondifferentiable, and we apply the majorize–minimize algorithm and estimate the asymptotic covariance using an efficient resampling method. We use simulation studies to evaluate the proposed method and illustrate its application by a real‐data example.     

Further results on the residual quantile entropy

It was shown that the decreasing mean residual life class implies the decreasing residual quantile entropy class and the decreasing residual quantile entropy class is not closed under formation of mixture. The less quantile entropy order was proved to be closed under the accelerated life models and the generalized order statistics models. Meanwhile, bounds of the entropy and the residual quantile entropy of some aging classes were established.     

Quantile residual life regression based on semi-competing risks data

This paper investigates the quantile residual life regression based on semi-competing risk data. Because the terminal event time dependently censors the non-terminal event time, the inference on the non-terminal event time is not available without extra assumption. Therefore, we assume that the non-terminal event time and the terminal event time follow an Archimedean copula. Then, we apply the inverse probability weight technique to construct an estimating equation of quantile residual life regression coefficients. But, the estimating equation may not be continuous in coefficients. Thus, we apply the generalized solution approach to overcome this problem. Since the variance estimation of the proposed estimator is difficult to obtain, we use the bootstrap resampling method to estimate it. From simulations, it shows the performance of the proposed method is well. Finally, we analyze the Bone Marrow Transplant data for illustrations.     

The estimation of treatment effect for the censored bivariate data under the univariate censoring

This paper establishes a nonparametric estimator for the treatment effect on censored bivariate data under unvariate censoring. This proposed estimator is based on the one from Lin and Ying(1993)'s nonparametric bivariate survival function estimator, which is itself a generalized version of Park and Park(1995)' quantile estimator. A Bahadur type representation of quantile functions were obtained from the marginal survival distribution estimator of Lin and Ying' model. The asymptotic property of this estimator is shown below and the simulation studies are also given     

The percentile residual life up to time t0: Ordering and aging properties

Motivated by practical issues, a new stochastic order for random variables is introduced by comparing all their percentile residual life functions until a certain instant. Some interpretations of these stochastic orders are given, and various properties of them are derived. The relationships to other stochastic orders are studied and also an application in reliability theory is described. Finally, we present some characterization results of the decreasing percentile residual life up to time t₀ aging notion.     

Tests for properties of residual life

The lOOα -percentile (0 < α < 1) residual life function at time x is defined to be the lOOα -percentile of the remaining life given survival up to time x. Joe and Proschan (1982b) develop tests for testing the alternatives representing decreasing 100α-percentile residual life (DPRL-α ) and the property ‘new better than used with respect to the lOOα -percentile’ (NBUP-α ). In this paper, tests are developed for DPRL[α, l) and NBUP[α, l) alternatives, where DPRL[α, l) is the class of life distributions which are DPRL-β distributions for all a ≤ β < 1 if 0 ≤ α < 1 and for all 0 < β < 1 if α = 0, and NBUP[α, l) is similarly defined. When α = 0, the DPRL[α, l) class of life distributions corresponds to the increasing failure rate class and the NBUP[α, l) class of life distributions corresponds to the new better than used class, and the test statistics are respectively asymptotically equivalent to the Hollander and Proschan (1975) test statistics for decreasing mean residual life and new better than used in expectation alternatives.     

On the Nonparametric Mean Residual Life Estimator in Length-biased Sampling

In this article, we discuss nonparametric estimation of a mean residual life function from length-biased data. Precisely, we prove strong uniform consistency and weak converge of the nonparametric mean residual life estimator in length-biased setting.     

Some reliability characteristics and stochastic ordering of series and parallel systems of bivariate generalized exponential distribution

In this paper, the reliability properties of two-component parallel and series systems are considered for bivariate generalized exponential (BVGE) distribution introduced by Kundu and Gupta [Bivariate generalized exponential distribution. J Multivar Anal. 2009;100:581–593]. For this model, the moments and mean residual life functions of these systems and the regression mean residual life function are derived. Stochastic comparisons of series and parallel systems of BVGE distribution are investigated. Moreover, various ordering results for the comparisons of series and parallel systems arising from BVGE random vectors are obtained with respect to the usual stochastic, reversed hazard rate and likelihood ratio orderings.     

Tests for bivariate mean residual life

In this paper we have developed tests for bivariate exponentiaIity against the ‘bivariate decreasing mean residual life (BDMRL)’ and ‘bivariate new better than used in expectation (BNBUE)’ classes of non-exponentia1 probability distributions. We have also obtained a large-sample approximation to make the test readily applicable.     

Residual life estimation based on bivariate Wiener degradation process with time-scale transformations

The issue of residual life (RL) estimation plays an important role for products while they are in use, especially for expensive and reliability-critical products. For many products, they may have two or more performance characteristics (PCs). Here, an adaptive method of RL estimation based on bivariate Wiener degradation process with time-scale transformations is presented. It is assumed that a product has two PCs, and that each PC is governed by a Wiener process with a time-scale transformation. The dependency of PCs is characterized by the Frank copula function. Parameters are estimated by using the Bayesian Markov chain Monte Carlo method. Once new degradation information is available, the RL is re-estimated in an adaptive manner. A numerical example about fatigue cracks is given to demonstrate the usefulness and validity of the proposed method.     

Some new stochastic orders based on quantile function

Recently, in the literature, the use of quantile functions in the place of distribution functions has provided new models, alternative methodology and easier algebraic manipulations. In this paper, we introduce new orders among the random variables in terms of their quantile functions like the reversed hazard quantile function, the reversed mean residual quantile function and the reversed variance residual quantile function orders. The relationships among the proposed orders and some existing orders are also discussed.     

On inference for mean residual life

In this paper, we propose a smooth nonparametric estimator of mean residual life based on a randomly censored sample. Large sample properties of the proposed estimator are examined. Also we study the asymptotic relative efficiency for different members in the family of test statistics, proposed by Lim and Park(1998), for testing whether or not the mean residual life changes its trend, and we discuss the efficiency values of loss due to censoring. Monte Carlo simulations are conducted to illustrate the performance of our estimation and investigate the performance of test statistics by the power of tests.     

Jackknifed kernel quantile estimators

A new quantile estimator is obtained by jackknifing the kernel quantile estimator. The asymptotic relative deficiency of the kernel quantile estimator relative to the jackknifed quantile estimator is investigated.     

A polar coordinate transformation for estimating bivariate survival functions with randomly censored and truncated data

This paper proposes a new estimator for bivariate distribution functions under random truncation and random censoring. The new method is based on a polar coordinate transformation, which enables us to transform a bivariate survival function to a univariate survival function. A consistent estimator for the transformed univariate function is proposed. Then the univariate estimator is transformed back to a bivariate estimator. The estimator converges weakly to a zero-mean Gaussian process with an easily estimated covariance function. Consistent truncation probability estimate is also provided. Numerical studies show that the distribution estimator and truncation probability estimator perform remarkably well.     

Estimation of percentile residual life function with left-truncated and right-censored data

This article focuses on the estimation of percentile residual life function with left-truncated and right-censored data. Asymptotic normality and a pointwise confidence interval that does not require estimating the unknown underlying distribution function of the proposed empirical estimator are obtained. Some simulation studies and a real data example are used to illustrate our results.     

Some new results on the cumulative residual entropy

The residual entropy function is a relevant dynamic measure of uncertainty in reliability and survival studies. Recently, Rao et al. [2004. Cumulative residual entropy: a new measure of information. IEEE Transactions on Information Theory 50, 1220–1228] and Asadi and Zohrevand [2007. On the dynamic cumulative residual entropy. Journal of Statistical Planning and Inference 137, 1931–1941] define the cumulative residual entropy and the dynamic cumulative residual entropy, respectively, as some new measures of uncertainty. They study some properties and applications of these measures showing how the cumulative residual entropy and the dynamic cumulative residual entropy are connected with the mean residual life function. In this paper, we obtain some new results on these functions. We also define and study the dynamic cumulative past entropy function. Some results are given connecting these measures of a lifetime distribution and that of the associated weighted distribution.     

Reliability functions of bivariate distributions with second kind of beta conditionals

The purpose of this article is to study the bivariate distributions with second kind of beta conditionals, presented by Castillo and Sarabia, from a reliability point of view. The reliability functions, viz the survival function, the failure rate and mean residual life function, are derived and their monotonicity is discussed for the univariate model as well as the conditional model. Finally, Clayton’s association measure is obtained in terms of the hazard gradient and its value, in the case of our model, is derived.     

Weighted extensions of generalized cumulative residual entropy and their applications

Abstract

In this paper, we consider weighted extensions of generalized cumulative residual entropy and its dynamic(residual) version. Our results include linear transformations, stochastic ordering, bounds, aging class properties and some relationships with other reliability concepts. We also define the conditional weighted generalized cumulative residual entropy and discuss some properties of its. For these concepts, we obtain some characterization results under some assumptions. Finally, we provide an estimator of the new information measure using empirical approach. In addition, we study large sample properties of this estimator.