New acceptance sampling plans based on life tests for Birnbaum–Saunders distributions

In this paper, we consider the Birnbaum–Saunders distribution as a life model to develop various acceptance sampling schemes based on the truncated life tests. We develop the double sampling plan and determine the design parameters satisfying both the producer's and consumer's risks simultaneously for the specified reliability levels in terms of the mean ratio to the specified life. We also propose a group sampling plan and determine the parameters by the above-mentioned two-point method. Tables are constructed for the proposed sampling plans and results are explained with examples.     

Bimodal Birnbaum–Saunders distribution with applications to non negative measurements

In this article, we introduce a new extension of the Birnbaum–Saunders (BS) distribution as a follow-up to the family of skew-flexible-normal distributions. This extension produces a family of BS distributions including densities that can be unimodal as well as bimodal. This flexibility is important in dealing with positive bimodal data, given the difficulties experienced by the use of mixtures of distributions. Some basic properties of the new distribution are studied including moments. Parameter estimation is approached by the method of moments and also by maximum likelihood, including a derivation of the Fisher information matrix. Three real data illustrations indicate satisfactory performance of the proposed model.     

Marshall–Olkin gamma–Weibull distribution with applications

Abstract

A Marshall–Olkin variant of the Provost type gamma–Weibull probability distribution is being introduced in this paper. Some of its statistical functions and numerical characteristics among others characteristics function, moment generalizing function, central moments of real order are derived in the computational series expansion form and various illustrative special cases are discussed. This density function is utilized to model two real data sets. The new distribution provides a better fit than related distributions as measured by the Anderson–Darling and Cramér–von Mises statistics. The proposed distribution could find applications for instance in the physical and biological sciences, hydrology, medicine, meteorology, engineering, etc.     

An alpha-power extension for the Birnbaum–Saunders distribution

Data on fatigue to exceed a critical value (or to grow to a critical level at which failure is likely to occur) is typically adjusted using the Birnbaum–Saunders (BS) distribution [see Birnbaum ZW, Saunders SC. A new family of life distributions. J Appl Probab. 1969a;6:319–327]. Although this type of distribution is asymmetric, in some cases the degree of skewness and/or kurtosis are outside the distributional range allowed by the BS distribution. Thus, a more adequate distribution model for better adjusting such unexpected deviations is called for. With this in mind, the main object of this paper is to propose an extension of the BS distribution based on the asymmetric alpha-power family of distributions [see Pewsey A, Gómez HW, Bolfarine H. Likelihood-based inference for power distributions. Test. 2012;21(4):775–789]. This extension is called the exponentiated BS distribution. We expect that by replacing the normal distribution by such more general family, a more flexible BS family is obtained. Asymmetry in the alpha-power family is controlled by a shape parameter, which also presents a similar performance in the extended BS family. The paper presents the density function for the extended BS and derives closed-form expressions for moments. Estimation is dealt with by using maximum likelihood estimators. Large sample inference can be conducted by using the Fisher information matrix derived in the paper. Estimation performance is studied by using a small scale simulation study. Results of a real application illustrates the good performance of the proposed approach.     

Random number generators for the generalized Birnbaum–Saunders distribution

The generalized Birnbaum–Saunders distribution pertains to a class of lifetime models including both lighter and heavier tailed distributions. This model adapts well to lifetime data, even when outliers exist, and has other good theoretical properties and application perspectives. However, statistical inference tools may not exist in closed form for this model. Hence, simulation and numerical studies are needed, which require a random number generator. Three different ways to generate observations from this model are considered here. These generators are compared by utilizing a goodness-of-fit procedure as well as their effectiveness in predicting the true parameter values by using Monte Carlo simulations. This goodness-of-fit procedure may also be used as an estimation method. The quality of this estimation method is studied here. Finally, through a real data set, the generalized and classical Birnbaum–Saunders models are compared by using this estimation method.     

Bayesian inference for Birnbaum–Saunders distribution and its generalization

We present a Bayesian approach for parameter inference of the Birnbaum–Saunders distribution [Birnbaum ZW, Saunders SC. A new family of life distributions. J Appl Probab. 1969;6:319–327], as well as the generalized Birnbaum–Saunders distribution developed by Owen [A new three-parameter extension to the Birnbaum–Saunders distribution. IEEE Trans Reliab. 2006;55:475–479], in the presence of random right-censored data. To handle the instance of commonly occurred censored observations, we utilize the data augmentation technique [Tanner MA, Wong WH. The calculation of posterior distributions by data augmentation. J Amer Statist Assoc. 1987;82(398):528–540] to circumvent the arduous expressions involving the censored data in posterior inferences. Simulation studies are carried out to assess performance of these methods under different parameter values, with small and large sample sizes, as well as various degrees of censoring. Two real data are analysed for illustrative purpose.     

An improved method of estimation for the parameters of the Birnbaum–Saunders distribution

The Birnbaum–Saunders (BS) distribution is a positively skewed distribution, frequently used for analysing lifetime data. In this paper, we propose a simple method of estimation for the parameters of the two-parameter BS distribution by making use of some key properties of the distribution. Compared with the maximum likelihood estimators and the modified moment estimators, the proposed method has smaller bias, but having the same mean square errors as these two estimators. We also discuss some methods of construction of confidence intervals. The performance of the estimators is then assessed by means of Monte Carlo simulations. Finally, an example is used to illustrate the method of estimation developed here.     

Robust explicit estimation of the two-parameter Birnbaum–Saunders distribution

The two-parameter Birnbaum–Saunders distribution is widely applicable to model failure times of fatiguing materials. Its maximum-likelihood estimators (MLEs) are very sensitive to outliers and also have no closed-form expressions. This motivates us to develop some alternative estimators. In this paper, we develop two robust estimators, which are also explicit functions of sample observations and are thus easy to compute. We derive their breakdown points and carry out extensive Monte Carlo simulation experiments to compare the performance of all the estimators under consideration. It has been observed from the simulation results that the proposed estimators outperform in a manner that is approximately comparable with the MLEs, whereas they are far superior in the presence of data contamination that often occurs in practical situations. A simple bias-reduction technique is presented to reduce the bias of the recommended estimators. Finally, the practical application of the developed procedures is illustrated with a real-data example.     

An extended Birnbaum–Saunders distribution: Theory,estimation, and applications

Abstract

Birnbaum and Saunders (1969a Birnbaum, Z.W., Saunders, S.C. (1969a). A new family of life distributions. J. Appl. Probab. 6:319–327.[Crossref], [Web of Science ®] , [Google Scholar]) pioneered a lifetime model which is commonly used in reliability studies. Based on this distribution, a new model called the gamma Birnbaum–Saunders distribution is proposed for describing fatigue life data. Several properties of the new distribution including explicit expressions for the ordinary and incomplete moments, generating and quantile functions, mean deviations, density function of the order statistics, and their moments are derived. We discuss the method of maximum likelihood and a Bayesian approach to estimate the model parameters. The superiority of the new model is illustrated by means of three failure real data sets. We also propose a new extended regression model based on the logarithm of the new distribution. The last model can be very useful to the analysis of real data and provide more realistic fits than other special regression models.     

Failure–censored reliability sampling plans for the exponential distribution

In this paper we examine the failure-censored sampling plans for the two–parameter exponential distri- bution based on m random samples, each of size n. The suggested procedure is based on exact results and only the first failure time of each sample is needed. The values of the acceptability constant are also tabulated for selected values of p _α ¹ p _β ¹, α and β. Further, a comparison of the proposed sampling plans with ordinary sampling plans using a sample of size mn is made. When compared to ordinary sampling plans, the proposed plan has an advantage in terms of shorter test-time and a saving of resources.     

Bayes Analysis and Comparison of Accelerated Weibull and Accelerated Birnbaum–Saunders Models

Several models are proposed in the literature for modeling fatigue data resulting from materials subject to cyclic stress and strain. Accelerated Weibull and accelerated Birnbaum–Saunders distributions are most commonly used models. Whereas the accelerated Weibull model is easier compared to accelerated Birnbaum–Saunders, it fails to represent the situation equally well. The present article focuses on Bayes analysis of the two models and provides a comparison based on some important Bayesian tools. Model compatibility study using predictive simulation ideas is preceded by the said comparison. Throughout, the posterior simulations are carried out by Markov chain Monte Carlo procedure.     

Inference and optimal censoring schemes for progressively censored Birnbaum–Saunders distribution

The aim of this paper is twofold. First we discuss the maximum likelihood estimators of the unknown parameters of a two-parameter Birnbaum–Saunders distribution when the data are progressively Type-II censored. The maximum likelihood estimators are obtained using the EM algorithm by exploiting the property that the Birnbaum–Saunders distribution can be expressed as an equal mixture of an inverse Gaussian distribution and its reciprocal. From the proposed EM algorithm, the observed information matrix can be obtained quite easily, which can be used to construct the asymptotic confidence intervals. We perform the analysis of two real and one simulated data sets for illustrative purposes, and the performances are quite satisfactory. We further propose the use of different criteria to compare two different sampling schemes, and then find the optimal sampling scheme for a given criterion. It is observed that finding the optimal censoring scheme is a discrete optimization problem, and it is quite a computer intensive process. We examine one sub-optimal censoring scheme by restricting the choice of censoring schemes to one-step censoring schemes as suggested by Balakrishnan (2007), which can be obtained quite easily. We compare the performances of the sub-optimal censoring schemes with the optimal ones, and observe that the loss of information is quite insignificant.     

Inference for the Birnbaum–Saunders Lifetime Regression Model with Applications

The Birnbaum–Saunders distribution is a positively skewed distribution that is frequently used for analyzing lifetime data. Regression analysis is widely used in this context when some covariates are involved in the life-test. In this article, we discuss the maximum likelihood estimation of the model parameters and associated inference. We discuss the likelihood-ratio tests for some hypotheses of interest as well as some interval estimation methods. A Monte Carlo simulation study is then carried out to examine the performance of the proposed estimators and the interval estimation methods. Finally, some numerical data analyses are done for illustrating all the inferential methods developed here.     

Optimal design for the Bradley–Terry paired comparison model

Paired comparisons are a popular tool for questionnaires in psychological marketing research. The quality of the statistical analysis of the responses heavily depends on the design, i.e. the choice of the alternatives in the comparisons. In this paper we show that the structure of locally optimal designs changes substantially with the parameters in the underlying utility. This fact is illustrated by elementary examples, where the optimal designs can be completely characterized. As an alternative maximin efficient designs are proposed which perform well for all parameter settings. Research supported by grant Ho 1286 of the German Research Council (Deutsche Forschungsgemeinschaft).     

Inference for the common mean of several Birnbaum–Saunders populations

The Birnbaum–Saunders distribution is a widely used distribution in reliability applications to model failure times. For several samples from possible different Birnbaum–Saunders distributions, if their means can be considered as the same, it is of importance to make inference for the common mean. This paper presents procedures for interval estimation and hypothesis testing for the common mean of several Birnbaum–Saunders populations. The proposed approaches are hybrids between the generalized inference method and the large sample theory. Some simulation results are conducted to present the performance of the proposed approaches. The simulation results indicate that our proposed approaches perform well. Finally, the proposed approaches are applied to analyze a real example on the fatigue life of 6061-T6 aluminum coupons for illustration.     

Capability indices for Birnbaum–Saunders processes applied to electronic and food industries

Process capability indices (PCIs) are tools widely used by the industries to determine the quality of their products and the performance of their manufacturing processes. Classic versions of these indices were constructed for processes whose quality characteristics have a normal distribution. In practice, many of these characteristics do not follow this distribution. In such a case, the classic PCIs must be modified to take into account the non-normality. Ignoring the effect of this non-normality can lead to misinterpretation of the process capability and ill-advised business decisions. An asymmetric non-normal model that is receiving considerable attention due to its good properties is the Birnbaum–Saunders (BS) distribution. We propose, develop, implement and apply a methodology based on PCIs for BS processes considering estimation, parametric inference, bootstrap and optimization tools. This methodology is implemented in the statistical software {tt R}. A simulation study is conducted to evaluate its performance. Real-world case studies with applications for three data sets are carried out to illustrate its potentiality. One of these data sets was already published and is associated with the electronic industry, whereas the other two are unpublished and associated with the food industry.     

L-moments of the Birnbaum–Saunders distribution and its extreme value version: estimation,goodness of fit and application to earthquake data

Understanding patterns in the frequency of extreme natural events, such as earthquakes, is important as it helps in the prediction of their future occurrence and hence provides better civil protection. Distributions describing these events are known to be heavy tailed and positive skew making standard distributions unsuitable for modelling the frequency of such events. The Birnbaum–Saunders distribution and its extreme value version have been widely studied and applied due to their attractive properties. We derive L-moment equations for these distributions and propose novel methods for parameter estimation, goodness-of-fit assessment and model selection. A simulation study is conducted to evaluate the performance of the L-moment estimators, which is compared to that of the maximum likelihood estimators, demonstrating the superiority of the proposed methods. To illustrate these methods in a practical application, a data analysis of real-world earthquake magnitudes, obtained from the global centroid moment tensor catalogue during 1962–2015, is carried out. This application identifies the extreme value Birnbaum–Saunders distribution as a better model than classic extreme value distributions for describing seismic events.     

Estimation of parameters for a Birnbaum–Saunders regression model with censored data

Little work has been published on the analysis of censored data for the Birnbaum–Saunders distribution (BISA). In this article, we implement the EM algorithm to fit a regression model with censored data when the failure times follow the BISA. Three approaches to implement the E-Step of the EM algorithm are considered. In two of these implementations, the M-Step is attained by an iterative least-squares procedure. The algorithm is exemplified with a single explanatory variable in the model.     

A Comparison of Two Random Number Generators for the Birnbaum–Saunders Distribution

Abstract

The Birnbaum–Saunders distribution was developed to describe fatigue failure lifetimes, however, the distribution has been shown to be applicable for a variety of situations that frequently occur in the engineering sciences. In general, the distribution can be used for situations that involve stochastic wear–out failure. The distribution does not have an exponential family structure, and it is often necessary to use simulation methods to study the properties of statistical inference procedures for this distribution. Two random number generators for the Birnbaum–Saunders distribution have appeared in the literature. The purpose of this article is to present and compare these two random number generators to determine which is more efficient. It is shown that one of these generators is a special case of the other and is simpler and more efficient to use.