Life Testing Problems Research Articles

The NBRULC Reliability Class: Mathematical Theory and Goodness-of-Fit Testing with Applications to Asymmetric Censored and Uncensored Data

The majority of approaches proposed in the past few decades to solve life test problems have differed markedly from those used for closely related, yet broader, issues. Due to the complexity of data that are generated each day in many practical domains, as a result of the development of scales for rating the success or failure of reliability, a new domain of reliability has been created. This domain is referred to as life classes, where specific probability distributions are presented. In this study, it is shown that the use of the quality-of-fit technique to solve problems involving life testing makes sense, and produces simpler processes that are roughly equivalent or superior to those used in traditional procedures. They may also behave better in limited samples. This work investigates a novel quality-of-fit test statistic; it is based on an exponential transform and is compared to the best renewal used Laplace test in increasing convex ordering (NBRULC). Evidence for approach normality is provided. The calculated variables include powers, Pitman asymptotic effectiveness, and critical points. Methods on how to handle censored data were also studied. Our experiments have real-world applications in the fields of medicine and engineering.

Analysis of adaptive type-II progressively hybrid censoring with binomial removals

Adaptive Type-II progressive hybrid censoring scheme has quite popular in a life-testing problem and reliability analysis due to it ensures more efficiency of inference procedures and saves total testing time. In this paper, an adaptive Type-II progressive hybrid censored sampling with random removals is considered, where the removals of the survival units at each stage from a life-test follows a binomial distribution during the execution of the experiment. The classical and Bayesian approaches are used to obtain the point and interval estimates of the unknown parameters of Weibull distribution, when the lifetimes are collected under the proposed censoring scheme. Different Bayesian estimates relative to various balanced type loss functions are obtained. Due to the complexity of the proposed estimators, some numerical techniques are implemented to obtain them. Using Markov chain Monte Carlo methods, the different Bayes estimates and associate credible intervals are developed. Extensive simulation study is performed to examine the efficiency of the proposed estimators. To show the applicability of the proposed methods in a real-life scenario, two real data sets coming from clinical and engineering areas are analysed.

Properties of increasing odds rate distributions with a statistical application

We study the family of distributions characterised by an increasing odds rate (IOR), showing that this provides an alternative interpretation of the “adverse ageing” notion, comparable to the well known increasing hazard rate (IHR) model. We prove some preservation properties of this class under several transformations that are often considered in reliability and life testing problems, including formation of order statistics. Moreover, the IOR assumption enables the derivation of survival bounds and tolerance limits, extending the scope of applicability of some known results, which are based, for instance, on the IHR assumption. Finally, we propose a test for the IOR null hypothesis, establishing its consistency and providing a table of simulated p-values.

Bayesian Estimation of Transmuted Weibull Distribution under Different Loss Functions

In this article, we aim to estimate the parameters of the transmuted Weibull distribution (TWD) using Bayesian approach, as the Weibull distribution plays an important role in reliability engineering and life testing problems. Informative and non-informative priors under squared error loss function (SELF), precautionary loss function (PLF) and quadratic loss function (QLF) are assumed to estimate the scale, the shape and the transmuted parameter of the TWD. In addition to this, we also compute the Bayesian credible intervals (BCIs). To estimate parameters, we adopt Markov Chain Monte Carlo (MCMC) technique assuming uncensored and censored environments in terms of different sample sizes and censoring rates. The posterior risks, associated with each estimator are used to compare the performance of different estimators. Two real data sets are analyzed to illustrate the flexibility of the proposed distribution.

An efficient family of Chebyshev–Halley’s methods for system of nonlinear equations

We suggest a new high-order family of iterative schemes for obtaining the solutions of nonlinear systems. The present scheme is an improvisation and extension of classical Chebyshev–Halley family for nonlinear systems along with higher-order convergence than the original scheme. The main theorem verifies the theoretical convergence order of our scheme along with convergence properties. In order to demonstrate the suitability of our technique, we choose several real life and academic test problems namely, boundary value, Bratu’s 2D, Fisher’s problems and some nonlinear system of minimum order of $$150\times 150$$ of nonlinear equations, etc. Finally, we wind up on the ground of computational consequences that our iterative methods demonstrate better performance than the existing schemes with respect to absolute residual errors, the absolute errors among two consecutive estimations and stable computational convergence order.

Adversarial issues in reliability

Many reliability problems involve two or more agents with conflicting interests whose decisions affect the performance of the system at hand. Examples of such problems relevant in management practice abound and include acceptance sampling, life testing, software testing, optimal maintenance, reliability demonstration, warranties and insurance. Most earlier attempts in such problems have focused on game theoretic approaches based on Nash equilibria and related concepts. However, these require strong common knowledge assumptions which do not frequently hold in practice. We provide an alternative framework based on adversarial risk analysis to deal with such problems which avoids the strong common knowledge assumptions of game theory. We illustrate the framework through acceptance sampling and life testing problems.

Transmuted Kumaraswamy-G Family of Distributions for Modelling Reliability Data

Abstract This research introduced the new class of distributions, called the transmuted Kumaraswamy (TKw) G family for modelling life testing problems. The new extended family is obtained by using the quadratic rank transmutation map method, which possesses a bathtub shape for its hazard rate. Some special models in the new family are discussed. We studied some mathematical properties of the new family, including the ordinary moments, moment generating functions, probability weighted moment, quantile function, and formulated the pdf of rth order statistics. The TKw G-distribution parameters are estimated using the method of maximum likelihood. We illustrate the potentiality of the new family with two applications to the failure and service times of Aircraft windshield data.

Weighted Lomax distribution.

The Lomax distribution (Pareto Type-II) is widely applicable in reliability and life testing problems in engineering as well as in survival analysis as an alternative distribution. In this paper, Weighted Lomax distribution is proposed and studied. The density function and its behavior, moments, hazard and survival functions, mean residual life and reversed failure rate, extreme values distributions and order statistics are derived and studied. The parameters of this distribution are estimated by the method of moments and the maximum likelihood estimation method and the observed information matrix is derived. Moreover, simulation schemes are derived. Finally, an application of the model to a real data set is presented and compared with some other well-known distributions.

Transmuted new generalized Weibull distribution for lifetime modeling

The Weibull family of lifetime distributions play a fundamental role in reliability engineering and life testing problems. This paper investigates the potential usefulness of transmuted new generalized Weibull (TNGW) distribution for modeling lifetime data. This distribution is an important competitive model that contains twenty-three lifetime distributions as special cases. We can obtain the TNGW distribution using the quadratic rank transmutation map (QRTM) technique. We derive the analytical shapes of the density and hazard functions for graphical illustrations. In addition, we explore some mathematical properties of the TNGW model including expressions for the quantile function, moments, entropies, mean deviation, Bonferroni and Lorenz curves and the moments of order statistics. The method of maximum likelihood is used to estimate the model parameters. Finally the applicability of the TNGW model is presented using nicotine in cigarettes data for illustration.

A derivation of prediction intervals for gamma regression

ABSTRACTGamma regression (GR) is a member of generalized liner models and often used when the phenomenon under study is skewed and the mean is proportional to the standard deviation. It can find applications in several areas such as life-testing problems, forecasting cancer incidences, weather extremes and quality control. Also it is a natural candidate when modelling the variance and it has been increasingly used over the past decade. When appropriate, studies encouraged its use over log transformation, Lewis et al. [Examples of designed experiments with nonnormal responses. J Qual Technol. 2001;33:265–278] and Hardin and Hilbe [Generalized linear models and extensions, 2nd ed. Stata Press; 2007]. Diagnostic tools and various inferential procedures such as confidence intervals and hypothesis testing were developed for GR models. An exception is prediction intervals for a future value of the process understudy. They are rarely discussed despite of its apparent importance for a practitioner. In this study I describe a simple but yet an effective procedure to compute prediction intervals for Gamma models. Four real data sets coming from medical and industrial experiments are studied to demonstrate the implementations and the usefulness of the suggested intervals. Extensive simulations are performed to investigate their advertised coverage under a wide range of sample sizes and several model designs showing consistency and satisfactory performance even for small samples.

Analysis of generalized progressive hybrid censored competing risks data

In reliability analysis, it is quite common for the failure of any individual or item to be attributable to more than one cause. Moreover, observed data are often censored. Recently, progressive hybrid censoring schemes have become quite popular in life-testing problems and reliability analysis. However, a limitation of the progressive hybrid censoring scheme is that it cannot be applied when few failures occur before time T. Therefore, generalized progressive hybrid censoring schemes have been introduced. In this article, we derive the likelihood inference of the unknown parameters under the assumptions that the lifetime distributions of different causes are independent and exponentially distributed. We obtain the maximum likelihood estimators of the unknown parameters in exact forms. Asymptotic confidence intervals are also proposed. Bayes estimates and credible intervals of the unknown parameters are obtained under the assumption of gamma priors on the unknown parameters. Different methods are compared using Monte Carlo simulations. One real data set is analyzed for illustrative purposes.

English

Rider (1957) introduced the generalized Cauchy distribution. In this paper we will consider a truncated version of the generalized Cauchy distribution. One possible use of the model is in life testing problem when the domain of application is non negative. Here we will consider several distributional properties of the truncated generalized Cauchy distribution. Some characterizations of this distribution based on truncated moments, order statistics and record values, will be given. Keywords: Order statistics; Record values; Characterization; Hazard rate.

Distribution of the Number of Observations Greater Than the ith Dependent Progressively Type-II Censored Order Statistic and Its Use in Goodness-of-fit Testing

In a life-testing problem, it may be of interest to investigate the number of observations close to but greater than the median, minimum, or more generally, the ith progressively Type-II censored order statistic (PCOS-II). In this paper, we derive the probability mass and distribution functions of the number of observations greater than the ith PCOS-II for a system with identical components for the cases of independent as well as dependent components. The type of dependence considered among the component lifetimes is through an Archimedean copula. We also provide a goodness-of-fit method for determining the best copula model for a given PCOS-II. Finally, an example is provided to illustrate the results developed here.

Estimating the Entropy of a Weibull Distribution under Generalized Progressive Hybrid Censoring

Recently, progressive hybrid censoring schemes have become quite popular in a life-testing problem and reliability analysis. However, the limitation of the progressive hybrid censoring scheme is that it cannot be applied when few failures occur before time T. Therefore, a generalized progressive hybrid censoring scheme was introduced. In this paper, the estimation of the entropy of a two-parameter Weibull distribution based on the generalized progressively censored sample has been considered. The Bayes estimators for the entropy of the Weibull distribution based on the symmetric and asymmetric loss functions, such as the squared error, linex and general entropy loss functions, are provided. The Bayes estimators cannot be obtained explicitly, and Lindley’s approximation is used to obtain the Bayes estimators. Simulation experiments are performed to see the effectiveness of the different estimators. Finally, a real dataset has been analyzed for illustrative purposes.

Estimation of Parameters of a Mixed Failure Time Distribution Which is a Mixture of Degenerate (Degenerate at Zero) and I. G. Distribution

The problem of estimation of parameters of a mixture of degenerate (at zero) and exponential distribution is considered by Jayade and Prasad (1990). The sampling scheme proposed in it is extended in this paper to a mixture of degenerate and Inverse Gaussian distribution. The Inverse Gaussian distribution is very relevant for studying reliability and life-testing problems. The inverse Gaussian being the first passage time distribution for Wiener process makes it particularly appropriate for failure or reaction time data analysis.

Exact likelihood inference for an exponential parameter under generalized progressive hybrid censoring scheme

Recently, progressive hybrid censoring schemes have become quite popular in a life-testing problem and reliability analysis. However, the limitation of the progressive hybrid censoring scheme is that it cannot be applied when few failures occur before time T. In this article, we propose a generalized progressive hybrid censoring scheme, which allows us to observe a pre-specified number of failures. So, the certain number of failures and their survival times are provided all the time. We also derive the exact distribution of the maximum likelihood estimator (MLE) as well as exact confidence interval (CI) for the parameter of the exponential distribution under the generalized progressive hybrid censoring scheme. The results of simulation studies and real-life data analysis are included to illustrate the proposed method.

A Bivariate Gamma Distribution in Life Testing

A bivariate gamma distribution as a model in life testing problem has been proposed and various statistical properties are studied.

A note on attribute life-testing

We shall consider an attribute life-testing problem with replacement. The term 'attribute life-testing' refers to situation where the actual life of the failed items are not known but only the number of failires in given interval of time is available. In earlier papers life-testing problems of such naturefor the non-replacement case have been treated.

Acceptance sampling by variables with life-test objectives

Sampling plans by variables with special references to life-test problems based on Weidull distribution are obtained. The life-test is terminated at the r-th failure. The sampling plan is such that a lot is accepted with a reassigned(high)probability when the specified mean life can be established. OC functions of the plans are obtained and Producer's risk is also discussed. A method of obtaining minimum sample size has been proposed so as to obtain the require number of failures with a prescribed time limit with a given probability .

Study on Asymptotic Property of Additive-Accelerated Mean Regression Model

Recurrent event data is a kind of important incomplete data existed in survival analysis, biological medicine research, reliability life test and other practical problems. This paper presents an additive-accelerated mean regression model for multiple type recurrent events data, and gives the estimation methods of unknown parameter and non-parameter function. Specially, the asymptotic properties of parameters estimation are proved.