Testing Symmetry Based On Empirical Likelihood Of Correlation Coefficient Between The Density Function And Distribution Function

Symmetry is a qualitative feature in probability model,which plays an important role in mathematics as well as in statistics.Because symmetry is an significant assumption for many statistical models.For example,symmetry assumptions are essential in deriving point estimates or interval estimates of location parameters.In the regression problem,the linear regression model usual y needs to assume that the residual is normally distributed or symmetric about zero.In nonparametric statistics,symmetry is a key assumption for the validity of signed rank method[1,2].For example,the Wilcoxon signed-rank test[3] used for testing the differences between two samples with unknown symmetric distribution functions,and symmetry of the sample population is also required by the Mann–Whitney-type tests for location parameter.In order to objectively measure whether a probability density function is more or less symmetric than another probability density function,the mathematical quantification of symmetry is essential.This paper focuses on the general measure of the symmetry or asymmetry of the density function.In this paper,we propose a kth correlation coefficient between the density function and distribution function of a continuous variable as a general measure of symmetry and asymmetry.Our main work is as follows:(1)First of all,we propose a root-n moment-based estimator of the kth correlation coefficient and present its some asymptotic results.(2)And then,we using the empirical likelihood method(EL)to consider statistical inference of the kth correlation coefficient.We can know that the empirical likelihood method statistic is shown to be asymptotically a standard chisquared distribution.(3)Next,we consider a residual based estimator of the kth correlation coefficient for a parametric regression model to test whether the density function of the true model error is symmetric or not.We estimate the model error by residual and present the asymptotic results of the residual-based kth correlation coefficient estimator as well as constructing its EL-based confidence intervals.(4)After that,simulation studies are conducted to examine the performance of the proposed estimators.We find that 1 is not always the best.When the sample size is large and the unknown density function is symmetric,the numerical performance of 0.5 is usual y better than 1 for the construction of estimators and confidence intervals.Last,we use our proposed estimators to analyze the air quality dataset.