New Family Of Life Distributions Research Articles

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.

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.

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.

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.

Bootstrap-based improved estimators for the two-parameter Birnbaum–Saunders distribution

In this paper, we consider the two-parameter Birnbaum–Saunders distribution proposed by Birnbaum and Saunders [Birnbaum, Z.W. and Saunders, S.C., 1969, A new family of life distributions. Journal of Applied Probability, 6, 319–327], which is commonly used for modeling the lifetime of materials and equipment. We consider different strategies of bias correction of the maximum-likelihood estimators for the parameters that index the distribution via bootstrap (parametric and nonparametric). The numerical evidence favors a particular bootstrap estimator based on parametric resampling. Finally, an example with real data is presented and discussed.

A new family of life distributions for dependent data: Estimation

One of the biggest problems in reliability analysis is determining an appropriate distribution of life data. Therefore, this paper develops the estimation aspect of a family of life distributions obtained from spherical distributions. Additionally, a new family of life distributions is proposed for dependent life data, together with an optimization algorithm based on the simulated annealing method. This algorithm is very efficient for optimization purposes and does not require any manipulation of the log-likelihood functions for the distributions proposed in this study.

Improved statistical inference for the two-parameter Birnbaum–Saunders distribution

We develop nearly unbiased estimators for the two-parameter Birnbaum–Saunders distribution [Birnbaum, Z.W., Saunders, S.C., 1969a. A new family of life distributions. J. Appl. Probab. 6, 319–327], which is commonly used in reliability studies. We derive modified maximum likelihood estimators that are bias-free to second order. We also consider bootstrap-based bias correction. The numerical evidence we present favors three bias-adjusted estimators. Different interval estimation strategies are evaluated. Additionally, we derive a Bartlett correction that improves the finite-sample performance of the likelihood ratio test in finite samples.

A new family of life distributions based on the elliptically contoured distributions

In this article we propose a new family of life distributions, generated from an elliptically contoured distribution, in which the density and some of its properties are obtained. Explicit expressions for the density are found for a large number of specific elliptical distributions, such as Pearson type VII, t, Cauchy, Kotz type, normal, Bessel, Laplace and logistic.

Estimating the Shift to Wear-Out of Systems Having Exponential-Weibull Life Distributions

We develop a new family of life distributions, called the exponential-Weibull wear-out distributions, for systems whose failure rate function is a constant up to a change-point (wear-out point) and strictly increasing afterward. We derive properties of these wear-out distributions and develop a Bayes adaptive procedure for estimating the change point. Recursive formulas are given for determining the posterior probability that the change has occurred and its Bayes estimator. Results of numerical simulations are given to illustrate the properties of the adaptive procedure.

A new family of life distributions

SummaryA new two parameter family of life length distributions is presented which is derived from a model for fatigue. This derivation follows from considerations of renewal theory for the number of cycles needed to force a fatigue crack extension to exceed a critical value. Some closure properties of this family are given and some comparisons made with other families such as the lognormal which have been previously used in fatigue studies.

A new family of life distributions

Summary A new two parameter family of life length distributions is presented which is derived from a model for fatigue. This derivation follows from considerations of renewal theory for the number of cycles needed to force a fatigue crack extension to exceed a critical value. Some closure properties of this family are given and some comparisons made with other families such as the lognormal which have been previously used in fatigue studies.