International Journal of Reliability and Safety, 2018
In this paper we use the prediction-interval approach to construct acceptance criteria to determi... more In this paper we use the prediction-interval approach to construct acceptance criteria to determine whether or not certain batches of products are acceptable. The procedure is intended to protect both producers and consumers against highly defective lots and demonstrate that a required quality level is met with certain probability. The prediction interval approach is particularly useful to employ when the lifetime of the product represents the quality characteristic of interest. On the basis of a progressively censored sample from the Weibull lifetime distribution, the problem of constructing acceptance criteria by predicting a future lifetime based on an independent past sample of lifetimes from the same distribution is addressed in a Bayesian setting with a dependent bivariate prior. The Metropolis-within-Gibbs Sampler algorithm is used to obtain a sequence of draws from the posterior predictive distribution of future observations. This sequence is used to derive the prediction intervals based on which the lot acceptance criteria are determined. An example using real data is illustrated.
Pakistan Journal of Statistics and Operation Research
In this paper, based on an adaptive Type-II progressive censoring scheme, estimation of flexible ... more In this paper, based on an adaptive Type-II progressive censoring scheme, estimation of flexible Weibull extension-Burr XII distribution is discussed. Maximum likelihood estimation and asymptotic confidence intervals of the unknown parameters are obtained. The adaptive Metropolis (AM) method is applied to carry out a Bayesian estimation procedure under symmetric and asymmetric loss functions and calculate the credible intervals. A simulation study is carried out to assess the performance of the estimators. Finally, a real life data set is used for illustration purpose.
American Journal of Mathematical and Management Sciences
Abstract Random censoring is frequently utilized to analyze lifetime data in life-testing and rel... more Abstract Random censoring is frequently utilized to analyze lifetime data in life-testing and reliability experiments. However, no published work discussed parameter estimation and testing homogeneity under random censoring for the mixture of Birnbaum-Saunders (BS) distributions, which is a useful model for describing reliability data. In this paper, first the proof of the identifiability of a finite mixture of BS distributions with g components is developed using Laplace transform of BS distributions. Then based on random censoring, parameter estimation and testing homogeneity are discussed. An EM algorithm is proposed to obtain the maximum likelihood (ML) estimates of the mixture model parameters and an EM test is implemented to test for homogeneity in the mixture of BS distributions. A Monte Carlo simulation study is conducted to evaluate the performance of the ML estimates through the bias and mean squared error. Additionally, the performance of the EM test is investigated through its size and power which are calculated by Monte Carlo simulation. Finally, an example of aircraft Windshield data is used to illustrate the estimation and testing procedures.
This paper deals with testing the number of components in a Birnbaum-Saunders mixture model under... more This paper deals with testing the number of components in a Birnbaum-Saunders mixture model under randomly right censored data. We focus on two methods, one based on the modified likelihood ratio test and the other based on the shortcut of bootstrap test. Based on extensive Monte Carlo simulation studies, we evaluate and compare the performance of the proposed tests through their size and power. A power analysis provides guidance for researchers to examine the factors that affect the power of the proposed tests used in detecting the correct number of components in a Birnbaum-Saunders mixture model. Finally an example of aircraft Windshield data is used to illustrate the testing procedure.
Finite mixture models have been used in many fields of statistical analysis such as pattern recog... more Finite mixture models have been used in many fields of statistical analysis such as pattern recognition, clustering and survival analysis, and have been extensively applied in different scientific areas such as marketing, economics, medicine, genetics and social sciences. Introducing mixtures of new generalized lifetime distributions that exhibit important hazard shapes is a major field of research aiming at fitting and analyzing a wider variety of data sets. The main objective of this article is to present a full mathematical study of the properties of the new finite mixture of the three-parameter Weibull extension model, considered as a generalization of the standard Weibull distribution. The new proposed mixture model exhibits a bathtub-shaped hazard rate among other important shapes in reliability applications. We analytically prove the identifiability of the new mixture and investigate its mathematical properties and hazard rate function. Maximum likelihood estimation of the model parameters is considered. The Kolmogrov-Smirnov test statistic is used to fit two famous data sets from mechanical engineering to the proposed model, the Aarset data and the Meeker and Escobar datasets. Results show that the two-component version of the proposed mixture is a superior fit compared to various lifetime distributions, either one-component or two-component lifetime distributions. The new proposed mixture is a significant statistical tool to study lifetime data sets in numerous fields of study.
Pakistan Journal of Statistics and Operation Research
This paper is devoted to study a new four- parameter additive model. The newly suggested model is... more This paper is devoted to study a new four- parameter additive model. The newly suggested model is referred to as the flexible Weibull extension-Burr XII distribution. It is derived by considering a serial system with one component following a flexible Weibull extension distribution and another following a Burr XII distribution. The usefulness of the model stems from the flexibility of its failure rate which accommodates bathtub and modified bathtub among other risk patterns. These two patterns have been widely accepted in several fields, especially reliability and engineering fields. In addition, the importance of the new distribution is that it includes new sub-models which are not known in the literature. Some statistical properties of the proposed distribution such as quantile function, the mode, the rth moment, the moment generating function and the order statistics are discussed. Moreover, the method of maximum likelihood is used to estimate the parameters of the model. Also, t...
International Journal of Reliability and Safety, 2018
In this paper we use the prediction-interval approach to construct acceptance criteria to determi... more In this paper we use the prediction-interval approach to construct acceptance criteria to determine whether or not certain batches of products are acceptable. The procedure is intended to protect both producers and consumers against highly defective lots and demonstrate that a required quality level is met with certain probability. The prediction interval approach is particularly useful to employ when the lifetime of the product represents the quality characteristic of interest. On the basis of a progressively censored sample from the Weibull lifetime distribution, the problem of constructing acceptance criteria by predicting a future lifetime based on an independent past sample of lifetimes from the same distribution is addressed in a Bayesian setting with a dependent bivariate prior. The Metropolis-within-Gibbs Sampler algorithm is used to obtain a sequence of draws from the posterior predictive distribution of future observations. This sequence is used to derive the prediction intervals based on which the lot acceptance criteria are determined. An example using real data is illustrated.
Pakistan Journal of Statistics and Operation Research
In this paper, based on an adaptive Type-II progressive censoring scheme, estimation of flexible ... more In this paper, based on an adaptive Type-II progressive censoring scheme, estimation of flexible Weibull extension-Burr XII distribution is discussed. Maximum likelihood estimation and asymptotic confidence intervals of the unknown parameters are obtained. The adaptive Metropolis (AM) method is applied to carry out a Bayesian estimation procedure under symmetric and asymmetric loss functions and calculate the credible intervals. A simulation study is carried out to assess the performance of the estimators. Finally, a real life data set is used for illustration purpose.
American Journal of Mathematical and Management Sciences
Abstract Random censoring is frequently utilized to analyze lifetime data in life-testing and rel... more Abstract Random censoring is frequently utilized to analyze lifetime data in life-testing and reliability experiments. However, no published work discussed parameter estimation and testing homogeneity under random censoring for the mixture of Birnbaum-Saunders (BS) distributions, which is a useful model for describing reliability data. In this paper, first the proof of the identifiability of a finite mixture of BS distributions with g components is developed using Laplace transform of BS distributions. Then based on random censoring, parameter estimation and testing homogeneity are discussed. An EM algorithm is proposed to obtain the maximum likelihood (ML) estimates of the mixture model parameters and an EM test is implemented to test for homogeneity in the mixture of BS distributions. A Monte Carlo simulation study is conducted to evaluate the performance of the ML estimates through the bias and mean squared error. Additionally, the performance of the EM test is investigated through its size and power which are calculated by Monte Carlo simulation. Finally, an example of aircraft Windshield data is used to illustrate the estimation and testing procedures.
This paper deals with testing the number of components in a Birnbaum-Saunders mixture model under... more This paper deals with testing the number of components in a Birnbaum-Saunders mixture model under randomly right censored data. We focus on two methods, one based on the modified likelihood ratio test and the other based on the shortcut of bootstrap test. Based on extensive Monte Carlo simulation studies, we evaluate and compare the performance of the proposed tests through their size and power. A power analysis provides guidance for researchers to examine the factors that affect the power of the proposed tests used in detecting the correct number of components in a Birnbaum-Saunders mixture model. Finally an example of aircraft Windshield data is used to illustrate the testing procedure.
Finite mixture models have been used in many fields of statistical analysis such as pattern recog... more Finite mixture models have been used in many fields of statistical analysis such as pattern recognition, clustering and survival analysis, and have been extensively applied in different scientific areas such as marketing, economics, medicine, genetics and social sciences. Introducing mixtures of new generalized lifetime distributions that exhibit important hazard shapes is a major field of research aiming at fitting and analyzing a wider variety of data sets. The main objective of this article is to present a full mathematical study of the properties of the new finite mixture of the three-parameter Weibull extension model, considered as a generalization of the standard Weibull distribution. The new proposed mixture model exhibits a bathtub-shaped hazard rate among other important shapes in reliability applications. We analytically prove the identifiability of the new mixture and investigate its mathematical properties and hazard rate function. Maximum likelihood estimation of the model parameters is considered. The Kolmogrov-Smirnov test statistic is used to fit two famous data sets from mechanical engineering to the proposed model, the Aarset data and the Meeker and Escobar datasets. Results show that the two-component version of the proposed mixture is a superior fit compared to various lifetime distributions, either one-component or two-component lifetime distributions. The new proposed mixture is a significant statistical tool to study lifetime data sets in numerous fields of study.
Pakistan Journal of Statistics and Operation Research
This paper is devoted to study a new four- parameter additive model. The newly suggested model is... more This paper is devoted to study a new four- parameter additive model. The newly suggested model is referred to as the flexible Weibull extension-Burr XII distribution. It is derived by considering a serial system with one component following a flexible Weibull extension distribution and another following a Burr XII distribution. The usefulness of the model stems from the flexibility of its failure rate which accommodates bathtub and modified bathtub among other risk patterns. These two patterns have been widely accepted in several fields, especially reliability and engineering fields. In addition, the importance of the new distribution is that it includes new sub-models which are not known in the literature. Some statistical properties of the proposed distribution such as quantile function, the mode, the rth moment, the moment generating function and the order statistics are discussed. Moreover, the method of maximum likelihood is used to estimate the parameters of the model. Also, t...
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