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.     

Epsilon Birnbaum–Saunders distribution family: properties and inference

In this paper we introduce a new extension for the Birnbaum–Saunder distribution based on the family of the epsilon-skew-symmetric distributions studied in Arellano-Valle et al. (J Stat Plan Inference 128(2):427–443, 2005). The extension allows generating Birnbaun–Saunders type distributions able to deal with extreme or outlying observations (Dupuis and Mills, IEEE Trans Reliab 47:88–95, 1998). Basic properties such as moments and Fisher information matrix are also studied. Results of a real data application are reported illustrating good fitting properties of the proposed model.     

Multivariate generalized Birnbaum—Saunders kernel density estimators

In this article, we first propose the classical multivariate generalized Birnbaum–Saunders kernel estimator for probability density function estimation in the context of multivariate non negative data. Then, we apply two multiplicative bias correction (MBC) techniques for multivariate kernel density estimator. Some properties (bias, variance, and mean integrated squared error) of the corresponding estimators are also investigated. Finally, the performances of the classical and MBC estimators based on family of generalized Birnbaum–Saunders kernels are illustrated by a simulation study.     

The Poisson Birnbaum–Saunders model with long-term survivors

In this paper, we propose a cure rate survival model by assuming that the number of competing causes of the event of interest follows the Poisson distribution and the time to event has the Birnbaum–Saunders (BS) distribution. We define the Poisson BS distribution and provide two useful representations for its density function which facilitate to obtain some mathematical properties. Two closed-form expressions for the moments of the new distribution are given. We estimate the parameters of the model with cure rate using maximum likelihood. For different parameter settings, sample sizes and censoring percentages, several simulations are performed. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and present some ways to perform a global influence study. We analyse a real data set from the medical area.     

Bivariate log Birnbaum–Saunders distribution

Univariate Birnbaum–Saunders distribution has received a considerable amount of attention in recent years. Rieck and Nedelman (A log-linear model for the Birnbaum–Saunders distribution. Technometrics, 1991;33:51–60) introduced a log Birnbaum–Saunders distribution. The main aim of this paper is to introduce bivariate log Birnbaum–Saunders distribution. The proposed model is symmetric and it has five parameters. It can be obtained using Gaussian copula. Different properties can be obtained using copula structure. It is observed that the maximum likelihood estimators (MLEs) cannot be obtained explicitly. Two-dimensional profile likelihood approach may be adopted to compute the MLEs. We propose some alternative estimators also, which can be obtained quite conveniently. The analysis of one data set is performed for illustrative purposes. Finally, it is observed that this model can be used as a bivariate log-linear model, and its multivariate generalization is also quite straight forward.     

On scale-mixture Birnbaum–Saunders distributions

We present for the first time a justification on the basis of central limit theorems for the family of life distributions generated from scale-mixture of normals. This family was proposed by Balakrishnan et al. (2009) and can be used to accommodate unexpected observations for the usual Birnbaum–Saunders distribution generated from the normal one. The class of scale-mixture of normals includes normal, slash, Student-t, logistic, double-exponential, exponential power and many other distributions. We present a model for the crack extensions where the limiting distribution of total crack extensions is in the class of scale-mixture of normals.     

Multivariate Birnbaum–Saunders regression model

In this paper, we propose a multivariate log-linear Birnbaum–Saunders regression model. We discuss maximum-likelihood estimation of the model parameters and provide closed-form expressions for the score function and for Fisher's information matrix. Hypothesis testing is performed using approximations obtained from the asymptotic normality of the maximum-likelihood estimator. Some influence methods, such as the local influence and generalized leverage are discussed and the normal curvatures for studying local influence are derived under some perturbation schemes. Further, a test for the homogeneity of the shape parameter of the multivariate regression model is investigated. A real data set is presented for illustrative purposes.     

Comparison of several Birnbaum–Saunders distributions

Birnbaum–Saunders (BS) distribution is widely used in reliability applications to model failure times. For several samples from possible different BS distributions, to prevent wrong conclusions in any further analysis, it is of importance to accompany a formal comparison for characteristic quantities of the distributions, including mean, quantile and reliability function difference. To this end, two test statistics, which are respectively based on the exact generalized p-value approach and the Delta method, are proposed and their behaviours are investigated. Simulation studies are carried out to examine the size and power performance of the newly proposed statistics. An interesting phenomenon is that in the finite sample simulations we conduct, the Delta method-based test almost uniformly outperforms the generalized p-value-based test although its sampling null distribution is simulated by Monte Carlo method. This might suggest that the sampling null distribution of the Delta method-based test statistic would have a fast convergence to its limit. The tests are also applied to analyse a real example on the fatigue life of 6061-T6 aluminium coupons for illustration.     

Robust inference for Birnbaum–Saunders regressions

We propose a robust likelihood approach for the Birnbaum–Saunders regression model under model misspecification, which provides full likelihood inferences about regression parameters without knowing the true random mechanisms underlying the data. Monte Carlo simulation experiments and analysis of real data sets are carried out to illustrate the efficacy of the proposed robust methodology.     

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.     

Bayesian inference for the Birnbaum–Saunders nonlinear regression model

We develop a Bayesian analysis for the class of Birnbaum–Saunders nonlinear regression models introduced by Lemonte and Cordeiro (Comput Stat Data Anal 53:4441–4452, 2009). This regression model, which is based on the Birnbaum–Saunders distribution (Birnbaum and Saunders in J Appl Probab 6:319–327, 1969a), has been used successfully to model fatigue failure times. We have considered a Bayesian analysis under a normal-gamma prior. Due to the complexity of the model, Markov chain Monte Carlo methods are used to develop a Bayesian procedure for the considered model. We describe tools for model determination, which include the conditional predictive ordinate, the logarithm of the pseudo-marginal likelihood and the pseudo-Bayes factor. Additionally, case deletion influence diagnostics is developed for the joint posterior distribution based on the Kullback–Leibler divergence. Two empirical applications are considered in order to illustrate the developed procedures.     

Influence diagnostics in Birnbaum–Saunders nonlinear regression models

We consider the issue of assessing influence of observations in the class of Birnbaum–Saunders nonlinear regression models, which is useful in lifetime data analysis. Our results generalize those in Galea et al. [8 Galea, M., Leiva, V. and Paula, G. A. 2004. Influence diagnostics in log-Birnbaum–Saunders regression models. J. Appl. Stat., 31: 1049–1064. [Taylor & Francis Online], [Web of Science ®] [Google Scholar]] which are confined to Birnbaum–Saunders linear regression models. Some influence methods, such as the local influence, total local influence of an individual and generalized leverage are discussed. Additionally, the normal curvatures for studying local influence are derived under some perturbation schemes. We also give an application to a real fatigue data set.     

The Generalized Birnbaum–Saunders Distribution and Its Theory,Methodology, and Application

In this paper, we discuss the class of generalized Birnbaum–Saunders distributions, which is a very flexible family suitable for modeling lifetime data as it allows for different degrees of kurtosis and asymmetry and unimodality as well as bimodality. We describe the theoretical developments on this model including properties, transformations and related distributions, lifetime analysis, and shape analysis. We also discuss methods of inference based on uncensored and censored data, diagnostics methods, goodness-of-fit tests, and random number generation algorithms for the generalized Birnbaum–Saunders model. Finally, we present some illustrative examples and show that this distribution fits the data better than the classical Birnbaum–Saunders model.     

A robust multivariate Birnbaum–Saunders distribution: EM estimation

We propose here a robust multivariate extension of the bivariate Birnbaum–Saunders (BS) distribution derived by Kundu et al. [Bivariate Birnbaum–Saunders distribution and associated inference. J Multivariate Anal. 2010;101:113–125], based on scale mixtures of normal (SMN) distributions that are used for modelling symmetric data. This resulting multivariate BS-type distribution is an absolutely continuous distribution whose marginal and conditional distributions are of BS-type distribution of Balakrishnan et al. [Estimation in the Birnbaum–Saunders distribution based on scalemixture of normals and the EM algorithm. Stat Oper Res Trans. 2009;33:171–192]. Due to the complexity of the likelihood function, parameter estimation by direct maximization is very difficult to achieve. For this reason, we exploit the nice hierarchical representation of the proposed distribution to propose a fast and accurate EM algorithm for computing the maximum likelihood (ML) estimates of the model parameters. We then evaluate the finite-sample performance of the developed EM algorithm and the asymptotic properties of the ML estimates through empirical experiments. Finally, we illustrate the obtained results with a real data and display the robustness feature of the estimation procedure developed here.     

Estimation of the Parameters of the Birnbaum–Saunders Distribution

Some alternative estimators to the maximum likelihood estimators of the two parameters of the Birnbaum–Saunders distribution are proposed. Most have high efficiencies as measured by root mean square error and are robust to departure from the model as well as to outliers. In addition, the proposed estimators are easy to compute. Both complete and right-censored data are discussed. Simulation studies are provided to compare the performance of the estimators.     

Small-Sample Corrections for Score Tests in Birnbaum–Saunders Regressions

In this article, we deal with the issue of performing accurate small-sample inference in the Birnbaum–Saunders regression model, which can be useful for modeling lifetime or reliability data. We derive a Bartlett-type correction for the score test and numerically compare the corrected test with the usual score test and some other competitors.     

Multivariate Birnbaum–Saunders distribution: properties and associated inference

The univariate fatigue life distribution proposed by Birnbaum and Saunders [A new family of life distributions. J Appl Probab. 1969;6:319–327] has been used quite effectively to model times to failure for materials subject to fatigue and for modelling lifetime data and reliability problems. In this article, we introduce a Birnbaum–Saunders (BS) distribution in the multivariate setting. The new multivariate model arises in the context of conditionally specified distributions. The proposed multivariate model is an absolutely continuous distribution whose marginals are univariate BS distributions. General properties of the multivariate BS distribution are derived and the estimation of the unknown parameters by maximum likelihood is discussed. Further, the Fisher's information matrix is determined. Applications to real data of the proposed multivariate distribution are provided for illustrative purposes.     

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.