Statistical Analysis And Parameter Estimation Of The Three-parameter Generalized Birnbaum-Saunders Distribution Whose Kernel Is T Distribution

The Binrbaum-Saunders(BS)distribution was derived in 1969 as a lifetime model for fatigue failure of a cycling working specimen by Birnbaum and Saunders,who made further contributions to the statistic properties and estimation methods of their new distribution.Half a century later,Díaz-García and Leiva-Sánchez firstly extended BS distribution to a lifetime distribution family by replacing its standard normal kernel distribution with any generalized elliptical distribution.Meanwhile,after improving the original fatigue lifetime model,Owen managed to derive a reasonable three-parameter generalization of classical BS distribution.Latest years,due to its flexibility and practical potential,generalized BS distribution becomes the focus again in the field of parameter statistics.This article starts with most generalized definition of BS distribution family of which the CDF is given the form as any multiple parameterized monotone function compounded within a central symmetric kernel distribution function.With researching this generalized form,the paper manages to establish an instructive theory on dealing with some crucial problems of generalized BS distribution including its graphic of density function and failure rate function,statistic properties and maximum likelihood estimation.Article gives a decent study,both theoretical and simulative,on two types of the three-parameter generalized student's t-distribution kernel BS distribution(GGBS-t)as a typical instance for confirming the usefulness of these analytic theories.First chapter is a historical review on the development of BS distribution.Second and third chapter focus on the method of analyzing monotonic interval of density function and failure rate function.The paper deduces a closed form conclusion for the monotonic interval of density function of GGBS-t distribution.Fourth chapter proves several statistic properties of GGBS-t distribution inherited from both BS distribution and student's tdistribution,in which the most essential is the nonexistence of moments.Fifth chapter respectively studies maximum likelihood estimation of Type-I and TypeII GGBS-t distribution,and simplifies the estimation equation by a new method,say quasimaximum likelihood estimation(QMLE),of which the essential is providing another consistent estimator for scale parameter in advance.Both the existence condition for QMLE estimator and the unicity of estimator under QMLE are proved in this chapter.Moreover,a theorem on how to find the invariant estimator for GBS distribution is clarified after finishing the theory of QMLE.A Monte-Carlo stimulation for the verification of validity of QMLE is provided following all the theoretical works.The paper is concluded by making use of the GGBS-t distribution model and QMLE method on analyzing Canadian climate data about wind speed distribution,practically enhancing the validity on the usefulness of GGBS-t distribution.