Bayesian Study of a Weibull–Dhillon Competing Risk Model and Its Reliability Implications via HMC Sampling Methodology

This article overviews some foundation issues in reliability and argues that generally it does not make sense to use classical notions of the hazard rate (HR) and mean residual life (MRL) to describe the HR and MRL of a device (the HR and failure rate are interchangeable terms). It provides a new methodology, using engineering notions of HR and MRL, to describe the HR and MRL of a device. and discusses the relevance of these methods in comparison to existing methods. In practice, the engineering notions of HR and MRL can be more complicated and more difficult to compute than the classical notions of these functions. Nevertheless, one should adhere to the procedures in this paper insofar as feasible because, as the examples indicate, the engineering notions of these functions dramatically differ from the classical notions. The finite populations have specified joint distributions.

12 Mean residual life: Theory and applications

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.

A new class of distributions with increasing, decreasing, bathtub-shaped and unimodal hazard rate forms called generalized quadratic hazard rate-power series distribution is proposed. The new distribution is obtained by compounding the generalized quadratic hazard rate and power series distributions. This class of distributions contains several important distributions appeared in the literature, such as generalized quadratic hazard rate-geometric, -Poisson, -logarithmic, -binomial and -negative binomial distributions as special cases. We provide comprehensive mathematical properties of the new distribution. We obtain closed-form expressions for the density function, cumulative distribution function, survival and hazard rate functions, moments, mean residual life, mean past lifetime, order statistics and moments of order statistics; certain characterizations of the proposed distribution are presented as well. The special distributions are studied in some details. The maximum likelihood method is used to estimate the unknown parameters. We propose to use EM algorithm to compute the maximum likelihood estimators of the unknown parameters. It is observed that the proposed EM algorithm can be implemented very easily in practice. One data set has been analyzed for illustrative purposes. It is observed that the proposed model and the EM algorithm work quite well in practice.

To represent lifetime data, a new one-parameter lifetime distribution referred to as ''Sujit distribution'' has been developed. This model is a mixture of the gamma and exponential distributions. It has been discussed how the recommended distribution's moments, kurtosis, shape, hazard rate function, skewness, as well as mean residual functions are among its key characteristics. Maximum Likelihood Valuation has been utilized to evaluate the parameters. A real-world data set was used in the last session to determine the distribution's best fit.. KEYWORDS :Lifetime distribution, Mean residual life function, Hazard rate function, Moments.

In this paper, we introduce the generalized Gompertz-power series class of distributions which is obtained by compounding generalized Gompertz and power series distributions. This compounding procedure follows same way that was previously carried out by [25] and [3] in introducing the compound class of extended Weibull-power series distribution and the Weibull-geometric distribution, respectively. This distribution contains several lifetime models such as generalized Gompertz, generalized Gompertz-geometric, generalized Gompertz-poisson, generalized Gompertz-binomial distribution, and generalized Gompertzlogarithmic distribution as special cases. The hazard rate function of the new class of distributions can be increasing, decreasing and bathtub-shaped. We obtain several properties of this distribution such as its probability density function, Shannon entropy, its mean residual life and failure rate functions, quantiles and moments. The maximum likelihood estimation procedure via a EM-algorithm is presented, and sub-models of the distribution are studied in details.

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.

A two-parameter generalization of Sujatha distribution (AGSD), which includes Lindley distribution and Sujatha distribution as particular cases, has been proposed. It's important mathematical and statistical properties including its shape for varying values of parameters, moments, coefficient of variation, skewness, kurtosis, index of dispersion, hazard rate function, mean residual life function, stochastic ordering, mean deviations, Bonferroni and Lorenz curves, and stress-strength reliability have been discussed. Maximum likelihood estimation method has been discussed for estimating its parameters. AGSD provides better fit than Sujatha, Aradhana, Lindley and exponential distributions for modeling real lifetime data.Journal of Institute of Science and TechnologyVolume 22, Issue 1, July 2017, Page: 66-83

In the present paper, a new one parameter lifetime distribution named, “Adya distribution’ has been proposed for modeling lifetime data from engineering. Its various statistical properties including moments and moments based measures, hazard rate function, mean residual life function, stochastic ordering, deviations from the mean and the median, Bonferroni and Lorenz curves, and stress-strength reliability have been studied. Both the method of moment and the maximum likelihood estimation have been discussed for estimating the parameter of the proposed distribution. A numerical example has been presented to test the goodness of fit of the proposed distribution over other one parameter lifetime distributions available in statistical literature.

This article introduces a two-parameter exponentiated Teissier distribution. It is the main advantage of the distribution to have increasing, decreasing and bathtub shapes for its hazard rate function. The expressions of the ordinary moments, identifiability, quantiles, moments of order statistics, mean residual life function and entropy measure are derived. The skewness and kurtosis of the distribution are explored using the quantiles. In order to study two independent random variables, stress–strength reliability and stochastic orderings are discussed. Estimators based on likelihood, least squares, weighted least squares and product spacings are constructed for estimating the unknown parameters of the distribution. An algorithm is presented for random sample generation from the distribution. Simulation experiments are conducted to compare the performances of the considered estimators of the parameters and percentiles. Three sets of real data are fitted by using the proposed distribution over the competing distributions.

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.

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.

Ch. 7. Hazard measure and mean residual life orderings: A unified approach

Parameter estimation and remaining useful life prediction of lubricating oil with HMM

An overview is presented of some theoretical results concerning the mean residual life function used in reliability theory. An extension of the concept to the bivariate case is introduced, and the relationship between the reliability and mean residual life function is derived. The properties of the function and conditions for asymptotic exponentiality of component life lengths are discussed. >