Statistical Properties Of The Generalized Birnbaum-Saunders Distribution

Some damage to machinery and equipment and some organs of the structure of the lesion, the natural environment pollution, and many other problems are due to the outside world or their own some factors caused for a long time, rather than caused by external force instantaneously.Therefore to establish a model to measure the effect of these factors is of great research significance, under this needs,Birnbaum-Saunders distribution arises at the historic moment.So in this thesis, the properties of Birnbaum-Saunders distribution are introduced, including its expectation, variance, covariance, kurtosis and skewness corresponding expression and reasoning is presented.But along with the further research, Birnbaum-Saunders distribution is not enough to the actual application of various complex data to build a more perfect model, thus extending the Birnbaum-Saunders distribution has become a new research hotspot,thus Generalized Birnbaum-Saunders distribution finally appear in front of the world.In the learning process we found that Generalized Birnbaum-Saunders distribution is closely interconnected with elliptic distribution, so we will also be necessary to learn the definition of the elliptical distribution and the corresponding probability density function.So, in the second chapter, elliptical distribution are introduced in the form of a list.In the third chapter of the paper,we study the definition, properties, and the related distribution, lifetimelanalysis, parameter estimation of Generalized Birnbaum-Saunders Distribution in detail.Through the definition of Generalize-d Birnbaum-Saunders distribution,if the random variables follow elliptical distribution in R,then the corresponding Birnbaum-Saunders Distribution can be called Generalized Birnbaum-Saunders distribution;The properties of Generalized Birnbaum-Saunders distribution mainly include moment, skewness, kurtosis, and some properties of the transformations.The lifetime analysis of Generalized Birnbaum-Saunders distribution reflects its advantage compared with normal model in measuring the impact of the continuing cycle force on the life time, the corresponding risk function and survival function, mean residual life, the critical point of risk function have been given;Parameter estimation of Generalized Birnbaum-Saunders Distribution mainly discusses the maximum likelihood estimation method and the modified moment estimation method, and the confidence interval of parameters is presented.