Statistical Inference Of Johnson S_B Distribution

Reliability is an engineering discipline developed in the middle of the 20th century.After more than half a century of development,reliability technology has made great achievements in theoretical research and fruitful results in engineering applications.At present,in statistics,product life is often assumed to be exponential distribution,Weibull distribution,lognormal distribution,gamma distribution,etc.The Johnson S_B distributions could to convert bounded variables to normal distributions,and their application in the field of reliability needs to be further investigated.Based on the above background,this paper researches the Johnson S_Bdistribution parameters,the corresponding stress-strength model,and the model selection problem.First,based on the two-parameter Johnson S_B distribution,this paper studies point estimation of parameters and interval estimation.In the statistical inference of point estimation,this paper presents the explicit expression of maximum likelihood estimation（MLE）and the uniform minimum variance unbiased estimator（UMVUE）of the distribution parameters.In interval estimation,this paper obtains the corresponding generalized confidence interval（GCI）by constructing the generalized pivot quantities（GPQs）of the distribution parameters,quantiles,and reliability.By Monte Carlo simulation method,the generalized confidence interval proposed in this paper is compared with the Wald interval estimation,and the results show that coverage probabilities of the GCIs close to the nominal level.Finally,it is illustrated by real data.Secondly,this paper applies the Johnson S_B distribution to the stress-strength model,and constructs a lower confidence limit estimator for the reliability of the model.In this paper,the interval estimation of the reliability is constructed by generalized inference,and the lower confidence limit of the interval is modified by the Fisher-Z method.The method proposed in this paper is compared with the Wald method.Through Monte Carlo simulation,it is found that the generalized lower confidence limit calculated by the proposed generalized method is closer to the nominal level,and the performance is obviously better.Finally,it is explained through a real example.Finally,since the Johnson S_B distribution has the same definition domain as the Beta distribution and the Kumaraswamy distribution,this paper selects the models of these three distributions through hypothesis testing.Firstly,assuming that Johnson S_B is the null hypothesis,the Beta distribution is the alternative hypothesis,and then the corresponding likelihood ratio statistic（RML）is constructed.The model selection is made according to the estimation of RML,and the mean and variance of the asymptotic distribution of RML are determined based on the central limit theorem.Statistical inference is performed,whereby the probability PCS_ASof choosing the correct model is calculated.Then,compare the PCS_MC with the probability of selecting the correct model under the large sample and the PCS_ASobtained by the simulation.Finally,observing the convergence speed and the minimum sample size（n^*）which respectively achieves the pre-given correct probability of selecting the correct probability（PCS^*）.Then assume that the Beta distribution is the null hypothesis and the Johnson S_B distribution is the alternative hypothesis,and similar statistical inferences are made.The same goes for the statistical inference process for the Johnson S_B distribution and the Kumaraswamy distribution.At last,examples are given.