Residual Lifetime Distributions Research Articles

Extreme Values and Reliability Measures of Continuous Erlang Mixtures

This study delves into Extreme Value Theory, focusing on the limiting distributions of continuous Erlang mixtures and their mixing distributions. It categorizes Erlang mixtures based on their extreme value types, demonstrating that the limiting distributions of the Erlang mixtures can be either Type I (Gumbel) or Type II (Frechet), contingent on the mixing distribution. Further, this study derives the mean residual lifetime and equilibrium distributions of these mixed distributions.

Robust assessment of life insurance products

AbstractIn this paper we propose a robust assessment for the premium of a standard life insurance contract with respect to the uncertainty on the estimated residual lifetime distribution function. Specifically, we provide a method to derive the range of values that the premium of a given contract can attain when considering all residual lifetime distribution functions that satisfy an $$L^2$$ L 2 distance constraint to a reference distribution function. Furthermore, we show that the $$L^2$$ L 2 distance constraint can be used as flexible starting point to include further information regarding future mortality.

Estimation of Rényi entropy for lifetime uncertainty

A system can be considered to be reliable if it operates successfully for a long period of time and has fewer uncertainties during its lifespan. In other words, lower the uncertainty of a random variable implies higher reliability [see Ebrahimi N. How to measure uncertaintyin the residual life time distribution. Sankhya: Indian J Stat SerA. 1996;58:48–57]. Motivated by this, the present study considers a generalized entropy function, namely the Rényi entropy in the quantile framework as a measure of uncertainty and proposes two nonparametric estimators for its computation. Asymptotic properties of the estimators are established under suitable regularity conditions. Simulation study and real data analysis are carried out to compare the performance and usefulness of the estimators.

Rényi Entropy for Past Lifetime Distributions with Application in Inactive Coherent Systems

In parallel with the concept of Rényi entropy for residual lifetime distributions, the Rényi entropy of inactivity time of lifetime distributions belonging to asymmetric distributions is a useful measure of independent interest. For a system that turns out to be inactive in time t, the past entropy is considered as an uncertainty measure for the past lifetime distribution. In this study, we consider a coherent system that includes n components and has the property that all the components of the system have failed at time t. To assess the predictability of the coherent system’s lifetime, we use the system’s signature to determine the Rényi entropy of its past lifetime. We study several analytical results, including expressions, bounds, and order properties for this measure.

Quantiles of the conditional residual lifetime

The study of the residual lifetime received considerable attention in survival analysis and in other disciplines like reliability theory and actuarial science. The quantile residual lifetime function, the inverse of the residual lifetime distribution , provides an interesting and well-studied measure to analyse residual lifetimes. In this paper we generalize the residual lifetime distribution and the quantile residual lifetime function by adding an extra conditioning of the form or , where is a second variable containing extra information on . We propose, for right-censored lifetimes, nonparametric estimators for this generalized conditional remaining lifetime distribution and the corresponding quantile function and we derive the asymptotic theory. In a simulation study, we show the good performance of the newly proposed quantile estimators and we discuss an application to real data on primary biliary cirrhosis.

A generalized Wiener process with dependent degradation rate and volatility and time-varying mean-to-variance ratio

In degradation analysis, there exist two natural features for the degradation data. The first is the dependence of degradation rate and degradation volatility, and the other is the time-varying mean-to-variance ratio. Ignoring them may lead to a significant bias in assessing lifetime information of materials. This paper proposes a generalized Wiener degradation model, which puts these two characteristics and unit-to-unit variation into consideration simultaneously. The proposed model includes many existing Wiener degradation models as special cases. Then, a generalized closed-form approximated residual lifetime distribution is given for the proposed model. Statistical inference for the model parameters is conducted based on the expectation maximizing (EM) method, and two simple auxiliary frameworks are developed for the determination of initial values and the forms of the time-scaled functions. The developed methodologies are then illustrated and verified in a simulation study and two real data analysis.

Parameter Estimation for Composite Dynamical Systems Based on Sequential Order Statistics from Burr Type XII Mixture Distribution

Considering the impact of the heterogeneous conditions of the mixture baseline distribution on the parameter estimation of a composite dynamical system (CDS), we propose an approach to infer the model parameters and baseline survival function of CDS using the maximum likelihood estimation and Bayesian estimation methods. The power-trend hazard rate function and Burr type XII mixture distribution as the baseline distribution are used to characterize the changes of the residual lifetime distribution of surviving components. The Markov chain Monte Carlo approach via using a new Metropolis–Hastings within the Gibbs sampling algorithm is proposed to overcome the computation complexity when obtaining the Bayes estimates of model parameters. A numerical example is generated from the proposed CDS to analyze the proposed procedure. Monte Carlo simulations are conducted to investigate the performance of the proposed methods, and results show that the proposed Bayesian estimation method outperforms the maximum likelihood estimation method to obtain reliable estimates of the model parameters and baseline survival function in terms of the bias and mean square error.

An extension of the Cox–Czanner divergence measure to residual lifetime distributions with applications

A time-dependent divergence measure is proposed to compare the survival functions of two lifetime random variables. It is shown that the proposed measure ranges between and for the proportional hazards case has the metric properties. Several properties of the divergence measure are investigated, among others, it is shown that the divergence between two survival functions does not depend on time if and only if they follow the proportional hazards model. The measure is also examined for various other well-known survival models such as the proportional odds model. The estimation of the suggested divergence measure for survival data is also discussed, and the asymptotic normal distribution of the resulting estimator is established. The proposed estimation is evaluated via simulation and further employed to compare the effects of two treatment groups on the overall survival times of kidney cancer patients.

Inference on quantile residual life for length-biased survival data

Length biased data occurs when a prevalent sampling is used to recruit subject into a study that investigates the time from an initial event to a terminal event. Such data are usually left-truncated and right-censored. While there have been accurate and efficient methods to estimate the survival function, not much work have been done regarding estimation of residual life time distribution or the summary parameters such as the median and quantiles of the residual life. In this paper we propose two ways to estimate the quantiles of the residual life time at fixed time points accounting for the length biased and censored na- ture of the data. We provide the asymptotic properties of these estimators and investigate them through simulation studies. Considering that the variance of these estimators require density estimation, we suggest an alternate approach to obtain the confidence intervals for the residual function. We apply these methods to a breast cancer dataset from National Surgical Adjuvant Breast and Bowel Project (NSABP).

A NEW GENERALIZED VARENTROPY AND ITS PROPERTIES

The variance of Shannon information related to the random variable \(X\), which is called varentropy, is a measurement that indicates, how the information content of \(X\) is scattered around its entropy and explains its various applications in information theory, computer sciences, and statistics. In this paper, we introduce a new generalized varentropy based on the Tsallis entropy and also obtain some results and bounds for it. We compare the varentropy with the Tsallis varentropy. Moreover, we explain the Tsallis varentropy of the order statistics and analyse this concept in residual (past) lifetime distributions and then introduce two new classes of distributions by them.

On a Time Dependent Divergence Measure between Two Residual Lifetime Distributions

On a Time Dependent Divergence Measure between Two Residual Lifetime Distributions

On extropy of past lifetime distribution

Recently Qiu et al. (2017) have introduced residual extropy as measure of uncertainty in residual lifetime distributions analogues to residual entropy (1996). Also, they obtained some properties and applications of that. In this paper, we study the extropy to measure the uncertainty in a past lifetime distribution. This measure of uncertainty is called past extropy. Also it is showed a characterization result about the past extropy of largest order statistics.

On information-based residual lifetime in survival models with delayed failures

At many practical instances, the initiating point events (e.g., shocks) affect an object not immediately, but after some random delay. These models were studied in the literature only for the case when an initial shock process is Poisson. In our paper, we generalize these results to a meaningful case of the generalized Polya process (GPP) of initial shocks that was recently introduced in the literature. Distinct from the Poisson process, the GPP possesses the property of dependent increments, which makes it much more attractive in applications. We derive the distribution of the time to failure for a system subject to the GPP with delays. Our main focus, however, is on analysis of the corresponding residual lifetime distribution that depends now on the full history (information) of the initiating shock process.

Stochastic comparison of residual lifetime mixture models

In this paper, we consider stochastic comparisons of mixtures (F∗ and G∗) of residual life distributions (Fθ and Gθ, θ>0) arising out of different baseline distributions (F and G) and different mixing distributions (H1 and H2). Under certain conditions on F, G, H1 and H2, we establish that the distributions F∗ and G∗ are stochastically ordered. Further, we make stochastic comparison of mixture distribution F∗ with the distribution of residual lifetime at random time Θ1.

On Dynamic Generalized Measures of Inaccuracy

Generalized information measures play an important role in the measurement of uncertainty of certain random variables, where the standard practice of applying ordinary uncertainty measures fails to fit. Based on the Renyi entropy and its divergence, we propose a generalized measure of inaccuracy of order α(≠1)>0 between two residual and past lifetime distributions of a system. We study some important properties and characterizations of these measures. A numerical example is given to illustrate the usefulness of the proposed measure.

Damage and failure of SiC fiber tows during environment activated slow crack growth: Residual behavior and Strength-Probability-Time diagrams

The paper investigates the damage and failure of fiber tows induced by slow crack growth activated by environment and applied load. The investigation is based on calculation of residual strengths and stress-lifetimes using models of slow crack growth on filaments and residual tensile behavior of fiber tows. This approach allows an exploration of behaviors and phenomena that cannot be assessed experimentally owing to various practical difficulties related to temperature and variability. It was applied to SiC-based Hi-Nicalon and Hi-Nicalon S fibers. Effects of constant deformation or force loading modes at high temperature in air were examined. Damage and rupture of fiber tows were described using the statistical distributions of filament residual strengths and lifetimes, tow residual behavior and Strength-Probability-Time (SPT) diagrams for critical filaments. The predicted SPT diagrams were compared to available experimental results obtained on tows and 2D woven SiC/SiC. Introduction of the effects of failure of groups of filaments in the model shed light on the trends in experimental results. The analysis highlighted the superior resistance to slow crack growth of Hi Nicalon S over Hi Nicalon which was related to superior fast fracture strength and toughness.

Reliability-Based Analytic Models for Fatigue Lifetime Distribution Estimation of Series Mechanical Systems under Random Load considering Strength Degradation Path Dependence

Time-dependent statistical characteristics of load process and strength degradation process are critical to lifetime distribution of series mechanical systems. Conventional rain-flow counting method for lifetime distribution estimation has limited practical application due to its strict requirement for statistical properties of load process. Besides, dynamic interaction between load process and strength degradation process results in strength degradation path dependence (SDPD). SDPD and failure dependence of components jointly bring considerable difficulties in prediction of system lifetime distribution and system residual lifetime distribution. To address these problems, reliability-based analytic models for estimation of whole lifetime distribution and residual lifetime distribution of series mechanical systems under random load are developed in this paper, which take the time-dependent statistical parameters of load process and strength degradation process as the input of the models. Furthermore, SDPD and failure dependence of components are taken into account in the proposed models in an explicit mathematical expression. The results show that SDPD, failure dependence of components, and initial strength dispersion have significant influences on system lifetime distribution.

Some results on residual entropy of ranked set samples

In this paper, a number of results are presented for the residual and past entropies of ranked set sample. In addition, some properties are compared with their counterparts in the simple random sample of size n=2. Finally, a dynamic residual measure of inaccuracy associated with two residual lifetime distributions of ranked set sample and simple random sample is obtained.

Main problems of controlling power quality in active–adaptive mains

Problems of controlling power quality in active-adaptive mains and controlling power parameters are examined for mains with noise caused by voltage and current distortions. Ways to control voltage at transforming substations of 110–220/10–20 kV are relevant at present. Different approaches for different purposes regarding problems of controlling power are presented. Measuring and calculating procedures are examined to determine which is preferable. Examples in which these procedures can be used are presented. For this purpose, it is necessary to develop a scientific basis for creating a device for automating, controlling, and protecting smart electric circuits with algorithms for automated adaptation to topology and variation in mains parameters by simulating electric mains that operate with modern control technologies. It makes possible to investigate the active-adaptive impact to distributing mains parameters under different types of loading and its distance from the power source. The algorithm of OLTC control for transformers of 110–220/10–20 kV is presented. The idea of OLTC control is as follows: OLTC responses, considering its residual lifetime and switching distribution over time, should depend on load curve: during the hours within which the load curve is smooth, the OLTC “accumulates” a period of time in which the load is variable in time and the OLTC responds more often. As a result, the OLTC residual lifetime is used efficiently for the active adaptive system of voltage control.

On preservation under univariate weighted distributions

We derive some new results for preservation of various stochastic orders and aging classes under weighted distributions. The corresponding reversed preservation properties as straightforward conclusions of the obtained results for the direct preservation properties, are developed. Damage model of Rao, residual lifetime distribution, proportional hazards and proportional reversed hazards models are discussed as special weighted distributions to try some of our results.