The Generalized Birnbaum-saunders Distribution And Its Statistical Analysis

The Birnbaum-Saunders distribution is widely used in the study of fatigue life problems.In this paper,the generalized Birnbaum-Saunders distribution is generalized by adding a parameter to the BS distribution.Its application field is roughly the same as that of the BS distribution,but it has better fitting effect compared with the BS distribution in some special fields.The main work of this paper is to study and analyze the statistical properties of GBS distribution,to find a more suitable example application of GBS distribution,and to analyze and demonstrate.Firstly,this paper introduces the concept of GBS distribution,gives the expression of probability density function and failure rate function,and studies the law of function image changing with parameters.Then we study the statistical properties of the origin moment and the central moment of GBS distribution,mainly using the third kind of modified Bessel function and its properties to obtain the mathematical expressions of the origin moment and the central moment of GBS distribution.In particular,the expectation and variance of the GBS distribution are obtained,and the coefficient of variation,skewness and kurtosis are also obtained.By BS distribution can be expressed as the relationship between the two inverse gaussian distribution is derived GBS have similar properties.GBS distribution can be regarded as two mixed distribution of the inverse gaussian distribution,the nature of the combined GBS and inverse gaussian distribution,makes many properties of GBS distribution clearer,derivation process is greatly simplified.In addition,the point estimation and interval estimation of GBS distribution parameters are mainly studied by numerical method.Under the principle of maximum likelihood estimation,the point estimation of parameters is studied by using the conjugate gradient method and the EM algorithm,and the numerical simulation of the point estimation is done to analyze and compare the two calculation methods.The confidence interval of three parameters of GBS distribution is given by using the asymptotic normality of maximum likelihood estimation,and the method of estimating the interval is introduced in detail.In addition,this paper also makes a numerical simulation study on the confidence interval of GBS distribution,and obtains the interval coverage of each parameter at different confidence levels.Finally,the superiority of GBS distribution over BS distribution is demonstrated through two examples.The parameter estimation of GBS distribution and BS distribution is obtained from the data of two examples,and the probability density function and cumulative distribution function are drawn,and the AIC value and BIC value of the two distributions are obtained.By comprehensive comparison,it is concluded that GBS distribution is superior to BS distribution in data fitting in practical applications.