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Abstract: Depending on the type of distribution, estimation of parameters are not sometimes simple in practice. In particular, this is the case for Birnbaum-Saunders distribution (BS). In this article, we present four different methods for estimating the parameters of a BS distribution. First, a simple graphical technique, analogous to probability plotting, is used to estimate the parameters and check for goodness-of-fit of failure times following a Birnbaum-Saunders distribution. Then, the maximum likelihood estimators and a modification of the moment estimators of a two-parameter Birnbaum–Saunders distribution are discussed. Finally, The jackknife technique is considered as another method which is appropriate for the small sample size case. Monte Carlo simulation is also used to compare the performance of all these estimators.

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The two-parameter Birnbaum–Saunders (BS) distribution was originally proposed as a failure time distribution
for fatigue failure caused under cyclic loading. BS model is a positively skewed statistical distribution which has
received great attention in recent decades. Several extensions of this distribution with various degrees of skewness,
kurtosis and modality are considered. In particular, a generalized version of this model was derived based on symmetrical
distributions in the real line named the generalized BS (GBS) distribution. In this article, we propose a
new family of life distributions, generated from elliptically contoured distributions, and the density and some of its
properties are obtained. Explicit expressions for the density of a number of specific elliptical distributions, such as
Pearson type VII, t, Cauchy, Kotz type, normal, Laplace and logistic are found. Another generalization of the BS
distribution is also presented using skew-elliptical distribution which makes its symmetry more flexible. Finally,
some examples are provided to illustrate application of the distribution.