Generalized Birnbaum Saunders Distribution Research Articles

The Geometric Generalized Birnbaum–Saunders model with long-Term Survivors

Introduction: A cure rate survival model was developed based on the assumption that the number of competing reasons for the event of interest has the Geometric distribution and the time allocated to the event of interest follows the Generalized Birnbaum-Saunders distribution.
 Methods: The Geometric Generalized Birnbaum–Saunders distribution was defined and two useful representations were represented for its density function which contributes to the creation of some mathematical properties. Furthermore, the parameters of the model with cure rate were estimated by using the maximum likelihood method.
 Results: Several simulations were performed and a real data set was analyzed from the medical area for different sample sizes and censoring percentages.In the melanoma data set and regarding the AIC and SBC selection criteria, the Geometric Generalized Birnbaum–Saunders distribution model was preferred and was selected as the appropriate model in the present study.
 Conclusion: Geometric Generalized Birnbaum–Saunders distribution is a highly flexible lifetime model which allows for different degrees of Kurtosis and asymmetry.by considering the advantages of the Geometric Generalized Birnbaum–Saunders distribution model, the model can be implemented as an appropriate alternative to explain or predict the survival time for long-term individuals.

Classical and Bayesian inference for an extended generalized Birnbaum-Saunders distribution

Due to the successful modeling on fatigue lifetime in various fields, the generalized Birnbaum-Saunders (GBS) distribution has been recently explored and extended from different respects. In this article, we study a varying-stress accelerated life test for GBS model to induce an extended GBS (EGBS) distribution on product failure times. We outline the attributes of this highly flexible distribution, present the classical maximum likelihood estimation method, develop a novel goodness of fit technique, and propose a new inference in Bayesian framework. The performance of the methods are assessed through a simulation study under various scenarios of distribution settings and sample sizes. For illustrative purpose, we consider two numerical examples to display the accuracy and efficiency of the proposed Bayesian method over the likelihood-based procedure.

An alternative for Laplace Birnbaum-Saunders distribution

Abstract In this paper, we first propose a new general method to introduce various lifetime distributions by choosing an appropriate kernel distribution. They have some characteristics in common with the well-known Birnbaum-Saunders distribution. Then, we choose the triangular distribution as a kernel model and construct the new distribution. This distribution has its support on the positive real axis and consists of two-pieces. We show that the newly defined distribution is in fact a generalized Birnbaum-Saunders distribution. It is mathematically tractable for studying its theoretical properties in detail. Different methods of estimation of parameters are proposed. The existence and uniqueness problem of the maximum likelihood estimation method is discussed. The performances of the estimators are evaluated through simulation studies. A real data fitting which compares it with the ordinary Birnbaum-Saunders, Laplace Birnbaum-Saunders and other some generalized Birnbaum-Saunders distributions is also given.

Singular matrix variate Birnbaum-Saunders distribution under elliptical models

This work sets the matrix variate Birnbaum–Saunders theory in the context of singular distributions and elliptical models. The termed singular matrix variate generalized Birnbaum–Saunders distribution is obtained with respect the Hausdorff measure. Several basic properties and particular cases of this distribution are also derived.

STOCHASTIC ORDERING OF LIFETIMES OF PARALLEL AND SERIES SYSTEMS COMPRISING HETEROGENEOUS DEPENDENT COMPONENTS WITH GENERALIZED BIRNBAUM-SAUNDERS DISTRIBUTIONS

In this paper, we discuss stochastic orderings of lifetimes of two heterogeneous parallel and series systems with heterogeneous dependent components having generalized Birnbaum–Saunders distributions. The comparisons presented here are based on the vector majorization of parameters. The ordering results are established in some special cases for the generalized Birnbaum–Saunders distribution based on the multivariate elliptical, normal, t, logistic, and skew-normal kernels. Further, we use these results by considering Archimedean copulas to model the dependence structure among systems with generalized Birnbaum–Saunders components. These results have been used to derive some upper and lower bounds for survival functions of lifetimes of parallel and series systems.

Modeling right-skewed financial data streams: A likelihood inference based on the generalized Birnbaum–Saunders mixture model

Finite mixture models have recently been considered for analyzing positive support economical data streams with non-normal features. In this paper, a new mixture model based on the novel class of generalized Birnbaum–Saunders distributions is proposed to enhance strength and flexibility in modeling heterogeneous lifetime data. Some characteristics and properties of this mixture model are outlined. By presenting a convenient hierarchical representation, a mathematically elegant and computationally tractable EM-type algorithm is adopted for computing maximum likelihood estimates. Theoretical formulae of well-known risk measures referring to the class of generalized Birnbaum–Saunders distributions are derived. Finally, the utility of the postulated methodology is illustrated with some real-world data examples.

Correction to: Statistical Inference for Type-I Generalized Birnbaum–Saunders Distribution

The funding information was missing in the original version of this article. The correct funding information is given.

Statistical Inference for Type-I Generalized Birnbaum–Saunders Distribution

A new generalized Birnbaum–Saunders distribution (Type-I GBS) was presented in Owen (IEEE Trans Reliab 55:475–479, 2006) to model a lifetime of a product under cyclic stresses by using a long memory process on the crack extensions. This highly flexible model includes the original BS distribution as a special case and can be widely applied in fatigue studies. In this article, we present the relevant properties, parameter estimation, and hypothesis testing for the distribution. We explore the traditional maximum likelihood estimation approach, and propose a new inference method for the GBS-I distribution. An extensive simulation study is carried out to assess performance of the methods, and a real data is analyzed where it is shown that the GBS-I model with the proposed method provides an efficient estimation and achieves a better fit than the classic likelihood-based procedure.

Robust multivariate control charts based on Birnbaum–Saunders distributions

ABSTRACTMultivariate control charts are powerful and simple visual tools for monitoring the quality of a process. This multivariate monitoring is carried out by considering simultaneously several correlated quality characteristics and by determining whether these characteristics are in control or out of control. In this paper, we propose a robust methodology using multivariate quality control charts for subgroups based on generalized Birnbaum–Saunders distributions and an adapted Hotelling statistic. This methodology is constructed for Phases I and II of control charts. We estimate the corresponding parameters with the maximum likelihood method and use parametric bootstrapping to obtain the distribution of the adapted Hotelling statistic. In addition, we consider the Mahalanobis distance to detect multivariate outliers and use it to assess the adequacy of the distributional assumption. A Monte Carlo simulation study is conducted to evaluate the proposed methodology and to compare it with a standard methodology. This study reports the good performance of our methodology. An illustration with real-world air quality data of Santiago, Chile, is provided. This illustration shows that the methodology is useful for alerting early episodes of extreme air pollution, thus preventing adverse effects on human health.

Bayesian inference for Birnbaum–Saunders distribution and its generalization

ABSTRACTWe 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.

Revisit of relationships and models for the Birnbaum-Saunders and inverse-Gaussian distributions

The Birnbaum-Saunders distribution was derived in 1969 as a lifetime model for a specimen subjected to cyclic patterns of stresses and strains, and the ultimate failure of the specimen is assumed to be due to the growth of a dominant crack in the material. The inverse Gaussian distribution is used to describe the first passage time for a particle (moving with constant velocity) that is subject to linear Brownian motion. These two models have a rich history, and they have been shown to be much related. In this article, these two models will be reviewed and comparisons will be made. Specifically, two moment-ratio diagrams will be presented that gives insight to the reason of both distributions often achieve similar fits to experimental data. Next, a generalized Birnbaum-Saunders distribution, will be presented and several properties will be derived. In particular, it will be shown that this generalized model can be expressed as a mixture of inverse Gaussian-type random variables (similar to the two-parameter Birnbaum-Saunders model). Estimation of the parameters in the generalized Birnbaum-Saunders distribution will be discussed. Lastly, some conclusions from this investigation are presented.

The exponentiated generalized Birnbaum–Saunders distribution

A new distribution called the exponentiated generalized Birnbaum–Saunders distribution is proposed and studied. The new distribution is quite flexible to model survival data, reliability problems, fatigue life studies and hydrological data. It can have decreasing, increasing, upside-down bathtub, bathtub-shaped, increasing–decreasing–increasing and decreasing–increasing–decreasing hazard rate functions. We provide a comprehensive account of the mathematical properties of the new distribution and various structural quantities are derived. We discuss maximum likelihood estimation of the model parameters for uncensored and censored data. Two empirical applications of the new model to real data are presented for illustrative purposes.

Generalized multivariate Birnbaum–Saunders distributions and related inferential issues

Birnbaum and Saunders introduced in 1969 a two-parameter lifetime distribution which has been used quite successfully to model a wide variety of univariate positively skewed data. Diaz-Garcia and Leiva-Sanchez [8] proposed a generalized Birnbaum–Saunders distribution by using an elliptically symmetric distribution in place of the normal distribution. Recently, Kundu et al. [13] introduced a bivariate Birnbaum–Saunders distribution, based on a transformation of a bivariate normal distribution, and discussed its properties and associated inferential issues. In this paper, we construct a generalized multivariate Birnbaum–Saunders distribution, by using the multivariate elliptically symmetric distribution as a base kernel for the transformation instead of the multivariate normal distribution. Different properties of this distribution are obtained in the general case. Special emphasis is placed on statistical inference for two particular cases: (i) multivariate normal kernel and (ii) multivariate-t kernels. We use the maximized log-likelihood values for selecting the best kernel function. Finally, a data analysis is presented for illustrative purposes.

Testing hypotheses in the Birnbaum–Saunders distribution under type-II censored samples

The two-parameter Birnbaum–Saunders distribution has been used successfully to model fatigue failure times. Although censoring is typical in reliability and survival studies, little work has been published on the analysis of censored data for this distribution. In this paper, we address the issue of performing testing inference on the two parameters of the Birnbaum–Saunders distribution under type-II right censored samples. The likelihood ratio statistic and a recently proposed statistic, the gradient statistic, provide a convenient framework for statistical inference in such a case, since they do not require to obtain, estimate or invert an information matrix, which is an advantage in problems involving censored data. An extensive Monte Carlo simulation study is carried out in order to investigate and compare the finite sample performance of the likelihood ratio and the gradient tests. Our numerical results show evidence that the gradient test should be preferred. Further, we also consider the generalized Birnbaum–Saunders distribution under type-II right censored samples and present some Monte Carlo simulations for testing the parameters in this class of models using the likelihood ratio and gradient tests. Three empirical applications are presented.

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.

An extension of the generalized Birnbaum–Saunders distribution

In this paper we present an extension of the generalized Birnbaum–Saunders distribution family introduced in [Díaz-García, J.A., Leiva-Sánchez, V., 2005. A new family of life distributions based on the contoured elliptically distributions. Journal of Statistical Planning and Inference 128 (2), 445–457] with a view to make it even more flexible in terms of its kurtosis coefficient. Properties involving moments and asymmetry and kurtosis indexes are studied for some special members of this family such as the slash Birnbaum–Saunders and slash- t Birnbaum–Saunders. Simulation studies for some particular cases and a real data analysis are also reported, illustrating the usefulness of the extension considered.

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.

Lifetime analysis based on the generalized Birnbaum-Saunders distribution

In this paper, we consider a family of generalized Birnbaum-Saunders distributions and present a lifetime analysis based mainly on the hazard function of this model. In addition, we carry out maxim...

Lifetime analysis based on the generalized Birnbaum–Saunders distribution

In this paper, we consider a family of generalized Birnbaum–Saunders distributions and present a lifetime analysis based mainly on the hazard function of this model. In addition, we carry out maximum likelihood estimation by using an iterative algorithm, which produces robust estimates. Asymptotic inference is also presented. Next, the quality of the estimation method is examined by means of Monte Carlo simulations. We then provide a practical example to illustrate the obtained results. From this example and based on goodness-of-fit methods, we show that the GBS distribution results in a more appropriate model for modeling fatigue data than other models commonly used to model this type of data. Finally, we estimate the hazard function and the critical point of this function.

Acceptance Sampling Plans from Truncated Life Tests Based on the Generalized Birnbaum–Saunders Distribution

In this article, we develop acceptance sampling plans when the life test is truncated at a pre-fixed time. The minimum sample size necessary to ensure the specified median life is obtained by assuming that the lifetimes of the test units follow a generalized Birnbaum–Saunders distribution. The operating characteristic values of the sampling plans as well as producer's risk are presented. Two examples are also given to illustrate the procedure developed here, with one of them being based on a real data from software reliability.