The Birnbaum-Saunders regression model is commonly used in reliability studies. We address the issue of performing inference in this class of models when the number of observations is small. We show that the likelihood ratio test tends to be liberal when the sample size is small, and we obtain a correction factor which reduces the size distortion of the test. The correction makes the error rate of he test vanish faster as the sample size increases. The numerical results show that the modified test is more reliable in finite samples than the usual likelihood ratio test. We also present an empirical application.
Birnbaum-Saunders models have been widely used to model positively skewed data. In this paper, we introduce a bivariate Birnbaum-Saunders distribution which has the means as parameters. We present some properties of the univariate and bivariate Birnbaum-Saunders models. We discuss the maximum likelihood and modified moment estimation of the model parameters and associated inference. A simulation study is conducted to evaluate the performance of the maximum likelihood and modified moment estimators. The probability coverages of confidence intervals are also discussed. Finally, a real-world data…
In this paper, we discuss some theoretical results and properties of a discrete version of the Birnbaum-Saunders distribution. We present a proof of the unimodality of this model. Moreover, results on moments, quantile function, reliability and order statistics are also presented. In addition, we propose a regression model based on the discrete Birnbaum-Saunders distribution. The model parameters are estimated by the maximum likelihood method and a Monte Carlo study is performed to evaluate the performance of the estimators. Finally, we illustrate the proposed methodology with the use of real …
We introduce, for the first time, a new class of Birnbaum-Saunders nonlinear regression models potentially useful in lifetime data analysis. The class generalizes the regression model described by Rieck and Nedelman [1991, A log-linear model for the Birnbaum-Saunders distribution, Technometrics, 33, 51-60]. We discuss maximum likelihood estimation for the parameters of the model, and derive closed-form expressions for the second-order biases of these estimates. Our formulae are easily computed as ordinary linear regressions and are then used to define bias corrected maximum likelihood estimate…
The Birnbaum-Saunders distribution has been used quite effectively to model times to failure for materials subject to fatigue and for modeling lifetime data. In this paper we obtain asymptotic expansions, up to order $n^{-1/2}$ and under a sequence of Pitman alternatives, for the nonnull distribution functions of the likelihood ratio, Wald, score and gradient test statistics in the Birnbaum-Saunders regression model. The asymptotic distributions of all four statistics are obtained for testing a subset of regression parameters and for testing the shape parameter. Monte Carlo simulation is prese…
The length-biased Birnbaum-Saunders distribution is both useful and practical for environmental sciences. In this paper, we initially derive some new properties for the length-biased Birnbaum-Saunders distribution, showing that one of its parameters is the mode and that it is bimodal. We then introduce a new regression model based on this distribution. We implement use the maximum likelihood method for parameter estimation, approach interval estimation and consider three types of residuals. An elaborate Monte Carlo study is carried out for evaluating the performance of the likelihood-based est…
In this paper 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, the likelihood ratio test and its Bartlett-corrected version. Our simulation results suggest that the corrected test we propose is more reliable than the other tests.
The two-parameter Birnbaum-Saunders distribution has been used succesfully 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…
The modeling of personal accident insurance data has been a topic of extreme relevance in the insurance literature. This kind of data often exhibits positive skewness and heavy tails. In this work, we propose a new quantile regression model based on the scale-mixture Birnbaum-Saunders distribution for modeling personal accident insurance data. The maximum likelihood estimates of the model parameters are obtained via the EM algorithm. Two Monte Carlo simulation studies are performed using the R software. The first study aims to analyze the performances of the EM algorithm to obtain the maximum …
In this paper, a generalization for the Birnbaum Saunders distribution, which has been applied to the modelling of fatigue failure times and reliability studies, is considered. The maximum likelihood estimators and statistical inference for the distribution parameters are presented. Corresponding bivariate and multivariate distributions are proposed. The proposed distribution is applied to model real data sets.
The Birnbaum-Saunders distribution is a flexible and useful model which has been used in several fields. In this paper, a new bimodal version of this distribution based on the alpha-skew-normal distribution is established. We discuss some of its mathematical and inferential properties. We consider likelihood-based methods to estimate the model parameters. We carry out a Monte Carlo simulation study to evaluate the performance of the maximum likelihood estimators. For illustrative purposes, three real data sets are analyzed. The results indicated that the proposed model outperformed some existi…
The Birnbaum-Saunders distribution, also known as the fatigue-life distribution, is frequently used in reliability studies. We obtain adjustments to the Birnbaum--Saunders profile likelihood function. The modified versions of the likelihood function were obtained for both the shape and scale parameters, i.e., we take the shape parameter to be of interest and the scale parameter to be of nuisance, and then consider the situation in which the interest lies in performing inference on the scale parameter with the shape parameter entering the modeling in nuisance fashion. Modified profile maximum l…
This paper introduces a new four-parameter lifetime model called the Weibull Birnbaum-Saunders distribution. This new distribution represents a more flexible model for the lifetime data. Its failure rate function can be increasing, decreasing, upside-down bathtub shaped, bathtub-shaped or modified bathtub shaped depending on its parameters. Some structural properties of the proposed model are investigated including expansions for the cumulative and density functions, moments, generating function, mean deviations, order statistics and reliability. The maximum likelihood estimation method is use…
We address the issue of performing inference on the parameters that index a bimodal extension of the Birnbaum-Saunders distribution (BS). We show that maximum likelihood point estimation can be problematic since the standard nonlinear optimization algorithms may fail to converge. To deal with this problem, we penalize the log-likelihood function. The numerical evidence we present show that maximum likelihood estimation based on such penalized function is made considerably more reliable. We also consider hypothesis testing inference based on the penalized log-likelihood function. In particular,…
In this paper, we introduce the Birnbaum-Saunders power series class of distributions which is obtained by compounding Birnbaum-Saunders and power series distributions. The new class of distributions has as a particular case the two-parameter Birnbaum-Saunders distribution. The hazard rate function of the proposed class can be increasing and upside-down bathtub shaped. We provide important mathematical properties such as moments, order statistics, estimation of the parameters and inference for large sample. Three special cases of the new class are investigated with some details. We illustrate …
The univariate Birnbaum-Saunders distribution has been used quite effectively to model times to failure for materials subject to fatigue and for modeling lifetime data. In this article, we define a skewed version of the Birnbaum-Saunders distribution in the multivariate setting and derive several of its properties. The proposed skewed multivariate model is an absolutely continuous distribution whose marginals are univariate Birnbaum-Saunders distributions. Estimation of the parameters by maximum likelihood is discussed and the Fisher's information matrix is determined. A skewed bivariate versi…
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. [2004, Influence diagnostics in log-Birnbaum-Saunders regression models. Journal of Applied Statistics, 31, 1049-1064] 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 of local influence are derived unde…
The paper by Leiva et al. (2010) introduced a skewed version of the sinh-normal distribution, discussed some of its properties and characterized an extension of the Birnbaum-Saunders distribution associated with this distribution. In this paper, we introduce a skewed log-Birnbaum-Saunders regression model based on the skewed sinh-normal distribution. Some influence methods, such as the local influence and generalized leverage are presented. Additionally, we derived the normal curvatures of local influence under some perturbation schemes. An empirical application to a real data set is presented…
Birnbaum and Saunders introduced a two-parameter lifetime distribution to model fatigue life of a metal, subject to cyclic stress. Since then, extensive work has been done on this model providing different interpretations, constructions, generalizations, inferential methods, and extensions to bivariate, multivariate and matrix-variate cases. More than two hundred papers and one research monograph have already appeared describing all these aspects and developments. In this paper, we provide a detailed review of all these developments and at the same time indicate several open problems that coul…
The Birnbaum-Saunders distribution has been widely applied in several areas of science and although several methodologies related to this distribution have been proposed, the problem of determining the optimal sample size for estimating its mean has not yet been studied. For this purpose, we propose a methodology to determine the optimal sample size under a decision-theoretic approach. In this approach, we consider loss functions for point and interval inference. Finally, computational tools in the R language were developed to use in practice.
In this paper we introduce the zero-adjusted Birnbaum-Saunders regression model. This new model generalizes at least seven Birnbaum-Saunders regression models. The idea of this modeling is mixing a degenerate distribution at zero with a Birnbaum-Saunders distribution. Besides the capacity to account for excess zeros, the zero-adjusted Birnbaum-Saunders distribution additionally produces an attractive modeling structure to right-skewed data. In this model, the mean and precision parameter of the Birnbaum-Saunders distribution and the probability of zeros can be related to linear and/or non-line…
In Mombeni et al. (2019), Birnbaum-Saunders and Weibull kernel estimators were introduced for the estimation of cumulative distribution functions (c.d.f.s) supported on the half-line $[0,\infty)$. They were the first authors to use asymmetric kernels in the context of c.d.f. estimation. Their estimators were shown to perform better numerically than traditional methods such as the basic kernel method and the boundary modified version from Tenreiro (2013). In the present paper, we complement their study by introducing five new asymmetric kernel c.d.f. estimators, namely the Gamma, inverse Gamma,…
Mixture models have received a great deal of attention in statistics due to the wide range of applications found in recent years. This paper discusses a finite mixture model of Birnbaum- Saunders distributions with G components, as an important supplement of the work developed by Balakrishnan et al. (2011), who only considered two components. Our proposal enables the modeling of proper multimodal scenarios with greater flexibility, where the identifiability of the model with G components is proven and an EM-algorithm for the maximum likelihood (ML) estimation of the mixture parameters is devel…
The premise of independence among subjects in the same cluster/group often fails in practice, and models that rely on such untenable assumption can produce misleading results. To overcome this severe deficiency, we introduce a new regression model to handle overdispersed and correlated clustered counts. To account for correlation within clusters, we propose a Poisson regression model where the observations within the same cluster are driven by the same latent random effect that follows the Birnbaum-Saunders distribution with a parameter that controls the strength of dependence among the indivi…
This is an invited contribution to the discussion on Professor Deborah Mayo's paper, "On the Birnbaum argument for the strong likelihood principle," to appear in Statistical Science. Mayo clearly demonstrates that statistical methods violating the likelihood principle need not violate either the sufficiency or conditionality principle, thus refuting Birnbaum's claim. With the constraints of Birnbaum's theorem lifted, we revisit the foundations of statistical inference, focusing on some new foundational principles, the inferential model framework, and connections with sufficiency and conditioni…
Deborah Mayo claims to have refuted Birnbaum's argument that the Likelihood Principle is a logical consequence of the Sufficiency and Conditionality Principles. However, this claim fails because her interpretation of the Conditionality Principle is different from Birnbaum's. Birnbaum's proof cannot be so readily dismissed. [arXiv:1302.7021]
The paper by Mayo claims to provide a new clarification and critique of Birnbaum's argument for showing that sufficiency and conditionality principles imply the likelihood principle. However, much of the arguments go back to arguments made thirty to forty years ago. Also, the main contention in the paper, that Birnbaum's arguments are not valid, seems to rest on a misunderstanding. [arXiv:1302.7021]
We discuss Birnbaum's result, its relevance to statistical reasoning, Mayo's objections and the result in [Electron. J. Statist. 7 (2013) 2645-2655] that the proof of this result doesn't establish what is commonly believed. [arXiv:1302.7021]
Combining p-values from independent statistical tests is a popular approach to meta-analysis, particularly when the data underlying the tests are either no longer available or are difficult to combine. A diverse range of p-value combination methods appear in the literature, each with different statistical properties. Yet all too often the final choice used in a meta-analysis can appear arbitrary, as if all effort has been expended building the models that gave rise to the p-values. Birnbaum (1954) showed that any reasonable p-value combiner must be optimal against some alternative hypothesis. …
In this discussion we demonstrate that fiducial distributions provide a natural example of an inference paradigm that does not obey Strong Likelihood Principle while still satisfying the Weak Conditionality Principle. [arXiv:1302.7021]
Autoregressive conditional duration (ACD) models are primarily used to deal with data arising from times between two successive events. These models are usually specified in terms of a time-varying conditional mean or median duration. In this paper, we relax this assumption and consider a conditional quantile approach to facilitate the modeling of different percentiles. The proposed ACD quantile model is based on a skewed version of Birnbaum-Saunders distribution, which provides better fitting of the tails than the traditional Birnbaum-Saunders distribution, in addition to advancing the implem…
An essential component of inference based on familiar frequentist notions, such as $p$-values, significance and confidence levels, is the relevant sampling distribution. This feature results in violations of a principle known as the strong likelihood principle (SLP), the focus of this paper. In particular, if outcomes $\mathbf{x}^*$ and $\mathbf{y}^*$ from experiments $E_1$ and $E_2$ (both with unknown parameter $θ$) have different probability models $f_1(\cdot),f_2(\cdot)$, then even though $f_1(\mathbf{x}^*;θ)=cf_2(\mathbf{y}^*;θ)$ for all $θ$, outcomes $\mathbf{x}^*$ and $\mathbf{y}^*$ may …
