Statistical Inference Of P(Y<X) For Inverse Lomax Distribution

In recent years,research on reliability issues has been a hot topic.Reliability is an important indicator for measuring product quality,The reliability of a product not only affects the performance of the product,but also the social stability.With the de-velopment of science technology and the wide application of electronic products,the reliability analysis of the system becomes more and more important.In the past,re-liability studies mostly assume that the product’s life follows exponential distribution,normal distribution,and Pareto distribution and so on,among which the second Pareto distribution,also known as Lomax distribution,has been widely used in reliability and life test research and has been subject to the attention of domestic and foreign scholars.Although there are many scholars who have conducted related research on Lomax dis-tribution,few scholars have studied inverse Lomax distribution.This article has been inspired by this and studies the statistical inference of the reliability R in three cases,one sample with inverse Lomax distribution,two samples with equal scale parameters,and two samples with different scale parameters,respectively.In this paper,we first study the statistical inference of R=P(t0<X)when the ran-dom variable X follows the inverse Lomax distribution(where t0 is a constant).We obtain maximum likelihood estimates(R)of R by computing and the asymptotic distri-bution of R,where the asymptotic distribution of R can be used to construct approximate confidence interval of R.In addition,the bootstrap method is proposed for a small sam-ple size,then the bootstrap-p and bootstrap-t confidence intervals of R are obtained.Monte-Carlo numerical simulation method is used to compare and analyze the mean square error and bias of R,average length and average coverage rate of asymptotic con-fidence interval,bootstrap-p confidence interval and bootstrap-t confidence interval of R under different parameter combinations.Secondly,we consider the statistical inference of R=P(Y<X)under the two-sample case where the random variables X and Y both follow the inverse Lomax distri-bution of the same scale parameters.The difference compared with the single sample is that by transforming R,we derive the exact distribution of R and construct the exact confidence interval of R by using the exact distribution of R.In the same way,the av-erage length and average coverage rate of exact confidence interval of R are given by numerical simulation.And we compare the fitting effect of inverse Lomax model and Rayleigh model through a set of real data.Finally,we extend the statistical inference of R=P(Y<X)to the inverse Lomax distribution where the random variables X and Y follow different scale parameters.Due to the speciality and complexity of the expression of R,we analyze the tendency of the mean square error and the bias of R.average length and average coverage rate of asymptotic confidence interval,bootstrap-p confidence interval and bootstrap-t confi-dence interval of R under different parameter combinations and sample sizes by the method of graphing.