Bivariate mean residual life

In survival analysis interest lies in modeling data that describe the time to a particular event. Informative functions, namely the hazard function and mean residual life function, can be obtained from the model's distribution function. We focus on the mean residual life function which provides the expected remaining life given that the subject has survived (i.e., is event-free) up to a particular time. This function is of direct interest in reliability, medical, and actuarial fields. In addition to its practical interpretation, the mean residual life function characterizes the survival distribution. In terms of mean residual life function inference, there are two shortcomings present in the current literature. First off, the shape of the functional is often restricted, which forces the researcher to make an assumption that may not be appropriate. Secondly, in cases where the shape of the functional is not parametrically specified, full inference is not obtained. The aim of our research is to provide a modeling approach that yields full inference for the mean residual life function, and is not restrictive on the shape of the functional. In particular, we develop general Bayesian nonparametric modeling methods for inference for mean residual life functions built from a mixture model for the associated survival distribution. Although the prior model is not placed on the mean residual life function directly, our methods offer rich inference for the desired functional. We place a Dirichlet process mixture model on the survival function, and discuss the importance of careful kernel selection to ensure desirable properties for the mean residual life function. We advocate for a mixture model with a gamma kernel and dependent baseline distribution for the Dirichlet process prior. We extend our model to the regression setting by modeling the joint distribution for the survival response and random covariates. This approach provides a flexible method for obtaining inference for the regression functionals when the number of random covariates is small to moderate. We further extend our methods to the scenario where interest lies in comparison of survival between two experimental groups. Typically, we expect the range of survival in the two groups to be the same, but exhibiting different characteristics over that range. Here, we develop a dependent Dirichlet process prior for the mixing distributions having shared locations across the two groups and varying weights to incorporate dependency between populations and achieve richer inferential results. The final scenario we consider is the case in which the researcher believes two populations have ordered mean residual life functions. For such applications, a prior model that incorporates an ordering constraint on the mean residual life functions is attractive. We introduce a mixture of Erlang distributions with weights constructed using Dirichlet process priors that provides the mean residual life ordering result. We demonstrate the utility of our modeling methods through simulation and real data examples. In addition, we draw comparisons with both parametric and semiparametric models.

12 Mean residual life: Theory and applications

A two-parameter Quasi Lindley distribution (QLD), of which the Lindley distribution (LD) is a particular case, has been introduced. Its moments, failure rate function, mean residual life function and stochastic orderings have been discussed. It is found that the expressions for failure rate function, mean residual life function, and stochastic orderings of the QLD shows its flexibility over Lindley distribution and exponential distribution. Although, the QLD has two parameters, the expressions for coefficients of variation, skewness, and kurtosis depend upon only one parameter. The maximum likelihood method and the method of moments have been discussed for estimating its parameters. The distribution has been fitted to some data-sets to test its goodness of fit to which earlier the Lindley distribution has been fitted by others and it is found that to almost all these data-sets the QLD provides closer fits than those by the Lindley distribution. Key words: Lindley distribution, moments, failure rate function, mean residual life function, stochastic ordering, estimation of parameters, goodness of fit. .

In reliability studies, the expected additional life time given that a component has survived until time t is called the mean residual life function (MRLF). This MRLF determines the distribution function uniquely. In this paper an interpretation of MRLF in renewal theory is presented and some characterizations of the exponential distribution are obtained. Finally, considering the general MRLF, a method is developed for obtaining the mixing distribution when the original distribution is exponential. Some examples are discussed, in one of which Morrison’s (1978) result is obtained as a special case.

The mean residual life (MRL) function is one of the basic parameters of interest in survival analysis. In this paper, we develop three procedures based on modified versions of empirical likelihood (EL) to construct confidence intervals of the MRL function with length-biased data. The asymptotic results corresponding to the procedures have been established. The proposed methods exhibit better finite sample performance over other existing procedures, especially in small sample sizes. Simulations are conducted to compare coverage probabilities and the mean lengths of confidence intervals under different scenarios for the proposed methods and some existing methods. Two real data applications are provided to illustrate the methods of constructing confidence intervals.

Statistical dependence and stochastic orders are two common tools used in distribution theory or, broadly speaking, reliability theory. In the context of the present work, we draw attention to the fact that reliability practitioners usually assume that the components constituting a system work independently. However, there are instances where the components are found to be stochastically dependent following a certain multivariate distribution. This gives rise to some error in computation of aging functions. In this paper, we propose a method based on statistical dependence and stochastic orders to compute the error incurred due to the faulty assumption that components are independently working. Our method makes the task of error analysis readily comprehensible. Earlier, the authors working in this direction considered some bivariate distributions and inferred about over/under assessment (OA/UA) of aging functions in silos, viz., survival function (SF), hazard rate, mean residual life function. However, in our work, we propose that OA/UA in the SF is closely related to statistical dependence, and OA/UA in some specific aging functions can be inferred with the help of stochastic orders, assuming that the orders for some other class of aging functions are known.

Estimation of two ordered mean residual life functions

Analysis of lognormal survival data.

Despite the existence of various competing risk (CR) models with varied formulations, their practical implementations do not align with some present CR problems. This is because certain CR data necessitates constructing CR models that are based on hazard rate (HR) patterns associated with cause‐specific failure times. To circumvent this obstacle, this study proposes an easy‐to‐handle methodology that involves constructing a minimum variable following Weibull or Dhillon failures. The model is called the Weibull–Dhillon competing risk (WDCR) model. Compared to many existing literature, the proposed WDCR model can analyze failure times that exhibit increasing and bathtub HRs and effectively characterize failure times linked to inverted bathtub and modified bathtub HRs. We examined several reliability characteristics and demonstrated the relationship between the model's mean residual life and HR function. Due to its complicated joint posterior distribution, the Hamiltonian Monte Carlo (HMC) procedure is employed for Bayesian inference of WDCR model parameters and reliability features. We provide two illustrations to illustrate the proposed model's practical value. The outcomes were better than those of the other contenders, indicating a valuable approach to modeling different CR data, particularly in the context of reliability analyses or other quantitative studies.

Residual coefficient of variation and some characterization results

In this paper, we study unimodality conditions for distributions that describe markets with stochastic demand. Such conditions naturally emerge in the analysis of game-theoretic models of market competition (Cournot games) and supply chain coordination (Stackelberg games). We express the price elasticity of expected demand in terms of the mean residual life (MRL) function of the demand distribution and characterize optimal prices or equivalently, points of unitary elasticity, as fixed points of the MRL function. This leads to economic interpretable conditions on the demand distribution under which such fixed points exist and are unique. We find that markets with increasing price elasticity of expected demand that eventually become elastic correspond to distributions with decreasing generalized mean residual life (DGMRL) and finite second moment. DGMRL distributions strictly generalize the widely used increasing generalized failure rate (IGFR) distributions. We further elaborate on the relationship of the two classes, link their limiting behavior at infinity and examine moment and closure properties of DGMRL distributions that are important in economic applications. The DGMRL unimodality condition is useful in the analysis of optimal decisions under uncertainty in settings that are not covered by the widely-used IGFR condition; thus, it can be of broader interest to the game-theory and operations research literature.

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.

The purpose of this article is to study the bivariate distributions, proposed by Hassan and Lii[ 19 ], in modeling marked point processes, from the reliability point of view. The reliability functions, viz. the survival function, the failure rate, and the mean residual life function, are derived and their monotonicity is discussed for the marginals as well as the conditional models. The monotonicity of the failure rate, in the case of series system, is also investigated. Finally, Clayton's association measure is obtained in terms of the hazard gradient and its value, in the case of our model, is derived.

The mean residual life (MRL) function for a given random variable T is the expected remaining lifetime of T after a fixed time point t. It is of great interest in survival analysis, reliability, actuarial applications, duration modelling, etc. Liang, Shen, and He [‘Likelihood Ratio Inference for Mean Residual Life of Length-biased Random Variable’, Acta Mathematicae Applicatae Sinica, English Series, 32, 269–282] proposed empirical likelihood (EL) confidence intervals for the MRL based on length-biased right-censored data. However, their -2log(EL ratio) has a scaled chi-squared distribution. To avoid the estimation of the scale parameter in constructing confidence intervals, we propose a new empirical likelihood (NEL) based on i.i.d. representation of Kaplan–Meier weights involved in the estimating equation. We also develop the adjusted new empirical likelihood (ANEL) to improve the coverage probability for small samples. The performance of the NEL and the ANEL compared to the existing EL is demonstrated via simulations: the NEL-based and ANEL-based confidence intervals have better coverage accuracy than the EL-based confidence intervals. Finally, our methods are illustrated with a real data set.

Some new results on the cumulative residual entropy