Bootstrap Inference On The Location Parameter For The Skew-normal Population

As an important statistical indicator of population distribution,location parameters are often used to describe the distribution characteristics of real data.Many established researches often assume that the population is normally distributed,but the real data commonly tends to be skew distributed with unimodal,thick tail and asymmetrical characteristics.It will lack of robustness and reduce the statistical accuracy if that assumption is still used for statistical inference.In addition,most of the existing studies use direct parameters in the estimation of the skew-normal location parameters.However,using the normal method to study the skew-normal data can easily lead to a great deviation and even draw a wrong conclusion.In such case,the hypothesis testing and interval estimation problems of location parameter for skew-normal population are researched based on the Bootstrap approach and central limit theorem.Then the statistical inference on the location parameters of the skewnormal population under dependent samples are considered.Firstly,using the Bootstrap approach and central limit theorem,the Bootstrap test statistics and Bootstrap confidence intervals for location parameter of single population are constructed based on the methods of moment estimation and maximum likelihood estimation,respectively.Next,the Behrens-Fisher-type and interval estimation problems of two populations are discussed.Further,the Bootstrap test statistics and Bootstrap confidence intervals are constructed respectively when the scale parameter or skewness parameter is known under dependent samples.Then the Behrens-Fisher-type and interval estimation problems of two populations are studied.Finally,the above approaches are applied to the real data examples of leaf area index,red blood cell count in athletes,gross domestic product and stock closing price.The simulation results indicate that,when independent samples are used to study the hypothesis testing problem of the location parameter of single skew-normal population,the methods based on moment estimation and maximum likelihood estimation are slightly liberal in small sample sizes.However,with the increase of sample size,the actual levels of the above two approaches are close to the nominal significance level of 0.05.Considering the Behrens-Fisher-type problem of two populations,the method based on the moment estimator performs better than that based on maximum likelihood estimator.Furthermore,the Bootstrap approach and approximate approach have similar results when the scale parameters are known,while the former is better than the latter in terms of the Type I error probability when the location parameters are known.For the Behrens-Fisher-type problem,the Bootstrap approach is better than the approximate approach in most sample sizes and parameter settings whether the scale parameter is known or the skewness parameter is known,especially in small sample cases.In actual case analyses,the excellent statistical properties of the Bootstrap approach are verified.