Latent Failure Times Research Articles

Reliability estimation of s-out-of-k system with Kumaraswamy distribution based on partially constant stress accelerated life tests

In reliability theory, the reliability inference for s-out-of-k systems holds significant importance. In this paper, we explore the estimation of reliability for s-out-of-k systems based on partially constant stress accelerated life tests. Assume that the latent failure times of the components follow the Kumaraswamy distribution. Maximum likelihood estimates for the unknown parameters are established, and their uniqueness is demonstrated. In addition, confidence intervals for the unknown parameters are constructed using the covariance matrix. Confidence intervals for the reliability functions are determined by the Delta method, while Bootstrap intervals are provided for comparison purposes. Subsequently, Bayesian point and interval estimates based on MCMC techniques considering different loss functions are discussed. Lastly, we conduct an extensive simulation study and analyse one real data set, which reveals that the Bayesian approach yields the best results.

A Frailty Model for Semi-competing Risk Data with Applications to Colon Cancer

In semi-competing risks (which generalizes the competing risks scenario), a subject may experience both terminal and non-terminal events, usually dependent, where the event time to the intermediate non-terminal event (say, tumor recurrence in cancer studies) is subject to censoring by the terminal event (say, death), but not vice-versa. As an alternative to the latent failure time formulation of semi-competing risks with joint survival functions, here, we consider an illness-death (multistate) shared frailty framework, where the dependency between the terminal and non-terminal failure times is incorporated via the power variance frailty between the conditional transition rates that are assumed Markov. Inference is conducted via maximum likelihood. A simulation study is conducted to evaluate the finite sample performance of the model parameters. Finally, we compare and contrast our power variance frailty proposal to known alternatives via application to a colon cancer dataset. Relevant R code for implementation of our model is available in GitHub.

Parametric inferences using dependent competing risks data with partially observed failure causes from MOBK distribution under unified hybrid censoring

In this communication, various statistical inferential procedures for estimating unknown model parameters are investigated via utilizing partially observed dependent competing risks data under the unified hybrid censoring scheme when the latent failure times follow Marshall–Olkin bivariate Kumaraswamy distribution. The existence and uniqueness of the maximum likelihood estimators (MLEs) have been established. By using asymptotic normality property of MLE, the approximate confidence intervals have been constructed via observed Fisher information matrix. Moreover, Bayes estimates and the highest posterior density credible intervals have been computed under a highly flexible gamma-Dirichlet prior distribution by using Markov chain Monte Carlo technique. In addition, to compare the performance of proposed methods, a Monte Carlo simulation has been carried out. Finally, a real-life data set has been analysed to illustrate the operability and applicability of the methods considered.

Inference of improved adaptive progressively censored competing risks data for Weibull lifetime models

Recently, an improved adaptive Type-II progressive censoring scheme is proposed to ensure that the experimental time will not pass a pre-fixed time and ends the test after recording a pre-fixed number of failures. This paper studies the inference of the competing risks model from Weibull distribution under the improved adaptive progressive Type-II censoring. For this goal, we used the latent failure time model with Weibull lifetime distributions with common shape parameters. The point and interval estimation problems of parameters, reliability and hazard rate functions using the maximum likelihood and Bayesian estimation methods are considered. Moreover, making use of the asymptotic normality of classical estimators and delta method, approximate intervals are constructed via the observed Fisher information matrix. Following the assumption of independent gamma priors, the Bayes estimates of the scale parameters have closed expressions, but when the common shape parameter is unknown, the Bayes estimates cannot be formed explicitly. To solve this difficulty, we recommend using Markov chain Monte Carlo routine to compute the Bayes estimates and to construct credible intervals. A comprehensive Monte Carlo simulation is conducted to judge the behavior of the offered methods. Ultimately, analysis of electrodes data from the life-test of high-stress voltage endurance is provided to illustrate all proposed inferential procedures.

Inference for a general family of inverted exponentiated distributions under unified hybrid censoring with partially observed competing risks data

In this paper, statistical inference of a competing risks model is developed based on a unified hybrid censoring scheme when the latent failure times follow a general family of inverted exponentiated distributions, which covers a wide range of lifetime distributions. Point and interval estimation methods for estimating the model parameters based on maximum likelihood and Bayesian approaches under non-restricted and restricted parameter spaces are developed. The Bayes estimates are obtained under the squared error loss function with non-informative and informative prior distributions. The existence and uniqueness of the maximum likelihood estimates of the model parameters are proved. A Monte Carlo simulation study is carried out to evaluate the performance of the proposed estimation procedures. To illustrate the proposed inferential procedures, a real data analysis is provided. Finally, some concluding remarks and future research directions are presented.

Bayesian and non‐Bayesian inference of Weibull lifetime model based on partially observed competing risks data under unified hybrid censoring scheme

AbstractIn this paper, a competing risk model is analyzed based on unified hybrid censoring scheme (UHCS). It is assumed that the latent failure times follow Weibull distributions with a common shape and different scale parameters. The maximum likelihood estimates (MLEs) and approximate confidence intervals (ACIs) of the distributional parameters are obtained. Sufficient conditions under which the MLEs exist (uniquely) have been studied. Further, the Bayes estimates are obtained with respect to symmetric and asymmetric loss functions. The non‐informative and informative prior distributions are considered for computing the approximate Bayes estimates. Furthermore, the highest posterior density (HPD) credible intervals have been obtained by using Markov Chain Monte Carlo (MCMC) method with the Gibbs sampling technique. The coverage probabilities for each confidence interval are computed. Then, hypothesis testing has been implemented using likelihood ratio statistic. A Monte Carlo simulation study is carried out to compare the performance of the proposed estimates. Finally, two real data sets are provided to illustrate the estimates.

Likelihood Inference for Copula Models Based on Left-Truncated and Competing Risks Data from Field Studies

Survival and reliability analyses deal with incomplete failure time data, such as censored and truncated data. Recently, the classical left-truncation scheme was generalized to analyze “field data”, defined as samples collected within a fixed period. However, existing competing risks models dealing with left-truncated field data are not flexible enough. We propose copula-based competing risks models for latent failure times, permitting a flexible parametric form. We formulate maximum likelihood estimation methods under the Weibull, lognormal, and gamma distributions for the latent failure times. We conduct simulations to check the performance of the proposed methods. We finally give a real data example. We provide the R code to reproduce the simulations and data analysis results.

On partially observed competing risk model under generalized progressive hybrid censoring for Lomax distribution

ABSTRACT We study a competing risks model under the assumption that latent failure times follow family of Lomax distributions. We obtain various inferences for model parameters when causes of failure are partially known and lifetime data are observed using a generalized progressive hybrid censoring scheme. The existence and uniqueness properties of maximum likelihood estimators of unknown parameters are established. Bayes estimators and associated credible intervals are also obtained. In addition, various inferences for unknown parameters are derived under order-restricted shape parameters of Lomax distributions. Finally, a simulation study is conducted to evaluate the performance of the proposed estimates. A real data set is also analysed for illustration purposes.

Estimation of Stress-Strength Reliability for Multicomponent System with Rayleigh Data

Inference is investigated for a multicomponent stress-strength reliability (MSR) under Type-II censoring when the latent failure times follow two-parameter Rayleigh distribution. With a context that the lifetimes of the strength and stress variables have common location parameters, maximum likelihood estimator of MSR along with the existence and uniqueness is established. The associated approximate confidence interval is provided via the asymptotic distribution theory and delta method. Meanwhile, alternative generalized pivotal quantities-based point and confidence interval estimators are also constructed for MSR. More generally, when the lifetimes of strength and stress variables follow Rayleigh distributions with unequal location parameters, likelihood and generalized pivotal-based estimators are provided for MSR as well. In addition, to compare the equivalence of different strength and stress parameters, a likelihood ratio test is provided. Finally, simulation studies and a real data example are presented for illustration.

Inference of partially observed causes for failure of Lomax competing risks model under type-II generalized hybrid censoring scheme

Time-to- failure under different causes of failure is known as a competing risks model. Practice, competing risks data can be appeared in different applications such as engineering fields or biological and medical lifetime studies as well as other related areas. Also, the causes of failure, which are competing may be partially observed. In this paper, we adopted the competing risks model with partially observed causes of failure when the latent failure times follow Lomax life distribution under type-II generalized hybrid censoring scheme. The maximum likelihood estimators of the model parameters with the associated confidence intervals are discussed. Moreover, Bayes estimators under importance sampling procedure with probability credible intervals are developed. The results are discussed using both real and simulated data sets for illustration purposes. Finally, the Monte Carlo simulation experiments are performed to assess and compare the different proposed methods with some brief comments.

Inference for constant-stress Weibull competing risks model under generalized progressive hybrid censoring

In this paper, accelerated life test is considered when the latent failure times follow Weibull competing risks models. When both scale and shape parameters are nonconstant and affected by operating stress, inference is discussed for a constant-stress model under a generalized progressive hybrid censoring. Maximum likelihood estimates together with existence and uniqueness are established, and the approximate confidence intervals for unknown parameters are proposed based on the asymptotic theory. Further, when the distributions of failure causes have common shape or scale parameters, associated point and interval estimates are also proposed. In addition, to compare the equivalence of the parameters of different Weibull competing risks, likelihood ratio tests for interested hypotheses are presented for complementary. Finally, extensive numerical studies and a real-life example are provided for illustrations.

A semiparametric mixture model approach for regression analysis of partly interval-censored data with a cured subgroup.

Failure time data with a cured subgroup are frequently confronted in various scientific fields and many methods have been proposed for their analysis under right or interval censoring. However, a cure model approach does not seem to exist in the analysis of partly interval-censored data, which consist of both exactly observed and interval-censored observations on the failure time of interest. In this article, we propose a two-component mixture cure model approach for analyzing such type of data. We employ a logistic model to describe the cured probability and a proportional hazards model to model the latent failure time distribution for uncured subjects. We consider maximum likelihood estimation and develop a new expectation-maximization algorithm for its implementation. The asymptotic properties of the resulting estimators are established and the finite sample performance of the proposed method is examined through simulation studies. An application to a set of real data on childhood mortality in Nigeria is provided.

Inference for a general family of inverted exponentiated distributions with partially observed competing risks under generalized progressive hybrid censoring

In this paper, statistical inference for a competing risks model is discussed when latent failure times belong to a general family of inverted exponentiated distributions. Based on a generalized progressive hybrid censored data with partially observed failure causes, estimations for unknown parameters are presented under nonrestricted and restricted parameter cases from classic and Bayesian perspectives, respectively. The existence and uniqueness of maximum likelihood estimators of the unknown parameters are established, and the associated approximate confidence intervals are also constructed via Fisher information matrix. In sequel, the Bayes estimators and credible intervals of the parameters are also obtained as well. Finally, the performance of different estimators are evaluated using Monte Carlo simulations and a real data set is also analyzed for illustration.

Reliability inference for stress-strength model based on inverted exponential Rayleigh distribution under progressive Type-II censored data

In this paper, stress-strength model is studied for an inverted exponential Rayleigh distribution (IERD) when the latent failure times are progressively Type-II censored. When both strength and stress random variables follow common IERD scale parameters, the maximum likelihood estimate of stress-strength reliability (SSR) is established and the associated approximate confidence interval is also constructed using the asymptotic distribution theory and delta method. By constructing pivotal quantities, another alternative generalized estimates for SSR are also proposed for comparison. Moreover, when there are arbitrary strength and stress parameters, likelihood and generalized pivotal based estimates are also presented. In addition, testing problem is gave for comparing the equality of different strength and stress parameters. Finally, simulation study and a real data example are provided for illustration.

Inference for partially observed competing risks model for Kumaraswamy distribution under generalized progressive hybrid censoring

ABSTRACT In this paper, inference for a competing risks model is studied when latent failure times follow Kumaraswamy distribution and causes of failure are partially observed. Under generalized progressive hybrid censoring, existence and uniqueness of maximum likelihood estimators of model parameters are established. The confidence intervals are obtained by using asymptotic distribution theory. We further compute Bayes estimators along with credible intervals. In addition, inference is also discussed when there is order restricted shape parameters. The performance of all estimates is investigated using Monte-Carlo simulations. Finally, analysis of a real data set is presented for illustration purposes.

Competing Hazards Regression Parameter Estimation Under Different Informative Priors

In the analysis of survival data, cause specific quantities of competing risks get considerable attention as compared to latent failure time approach. This article focuses on parametric regression analysis of survival data using cause specific hazard function with Burr type XII distribution as a baseline model. We obtained maximum likelihood and Bayes estimates of cumulative cause specific hazard functions under competing risk setup. For Bayesian point of view we proposed a class of informative priors for parameters to observe the comprehensive compatibility and their effectiveness under two different loss functions. The appropriateness of model is measured by the simulation study. Finally, we illustrate the proposed methodologies using bone marrow transplant data from the Princess Margaret Hospital Ontario, Canada.

A Cure Rate Model for Exponentially Distributed Lifetimes with Competing Risks

In real life, more often experimental units are susceptible to more than one risk factor. Moreover, some experimental units may not fail even if they are observed over a long period of time. In statistical analysis, competing risks models handle the first scenario while cure rate models have been introduced to analyze the long-term survivors in the population. In this paper, we consider a cure rate model when the failure of a unit can be due to either of the two competing causes. To analyze the competing risk data in presence of long-term survivors, we consider the latent failure times approach introduced by Cox (J R Stat Soc Ser B (Methodol) 21(2):411–421, 1959). The latent failure times are assumed to follow exponential distributions, and they are independently distributed. Under this setup, a random censoring scheme is applied and the observed data consist of either censored times or actual failure times along with the cause of failures. We derive the maximum likelihood estimators (MLEs) using the expectation-maximization (EM) algorithm based on the missing value principle. As the overall survival function is not a proper survival function, the asymptotic behavior of the MLEs is not immediate. We provide the sufficient conditions for the existence, uniqueness, consistency and the asymptotic normality of the MLEs. Monte Carlo simulations are performed to support the theoretical validation numerically. For illustrative purposes, we have analyzed one real dataset, and the results are quite satisfactory.

Inference for dependence competing risks with partially observed failure causes from bivariate Gompertz distribution under generalized progressive hybrid censoring

AbstractCompeting risks model is considered with dependence causes of failure in this paper. When the latent failure times are distributed by a bivariate Gompertz model, statistical inference for the unknown model parameters is studied from classical and Bayesian approaches, respectively. Under a generalized progressive hybrid censoring, maximum likelihood estimators of the unknown parameters together with the associated existence and uniqueness are established, and the approximate confidence intervals are also obtained based on asymptotic likelihood theory via the observed Fisher information matrix. Moreover, Bayes estimates and the highest posterior density credible intervals of the unknown parameters are also provided based on a flexible Gamma–Dirichlet prior, and Monte Carlo sampling method is also derived to compute associated estimates. Finally, simulation studies and a real‐life example are given for illustration purposes.

Inference for dependent competing risks from bivariate Kumaraswamy distribution under generalized progressive hybrid censoring

In this paper, competing risks model is considered when causes of failure are dependent. When latent failure times are distributed by the Marshall-Olkin bivariate Kumaraswamy model, inference for the unknown model parameters is studied under a generalized progressive hybrid censoring. Maximum likelihood estimates of unknown parameters are established, and the associated existence and uniqueness are provided. The approximate confidence intervals are constructed via the observed Fisher information matrix. Moreover, Bayes estimates and the credible intervals of the unknown parameters are also presented based a flexible Gamma-Dirichlet prior, and the importance sampling method is used to compute associated estimates. Simulation study and a lifetime example are given for illustration purposes.

Inference for Weibull competing risks model with partially observed failure causes under generalized progressive hybrid censoring

In this paper, a competing risks model is studied when the latent failure times follow Weibull distribution. When the failure times are observed under generalized progressive hybrid censoring and the causes of failure are partially observed, the maximum likelihood estimators of the model parameters are established together with associated existence and uniqueness, and the approximate confidence intervals are constructed based on large sample theory. Bayes estimators and associated credible intervals are obtained under fairly general priors. Moreover, classical and Bayesian inferences are also discussed when there is an order restriction on the scale parameters of the Weibull distributions. Finally, a simulation study and a real data example are presented for illustration.