The Application Of Bernstein Polynomial Estimation In Multivariate Distribution And Grouped Data

In the application of statistics,the study of grouped data is one of the important topics that has attracted the attention of scholars.It is widely used in many fields such as actuarial science,medicine,biological science,agronomy and so on.Many scholars are interested in the density estimation of grouped data,and put forward many parametric statistics and nonparametric statistics to estimate the density and distribution function based on grouped data.In nonparametric statistics,the kernel density estimation method is often used,but it will be affected by the boundary effect,which limits the accuracy of the estimation results.Multivariate density estimation is one of the most difficult problems in nonparametric statistics.The empirical distribution and kernel density method are often adopted for multivariate density estimation,but the empirical distribution method is not a smooth approximation,and will produce large deviation when the number of samples is small,and the estimation effect is poor.Therefore,the research on grouped data and multivariate distribution has theoretical value.Since the Bernstein polynomial was proposed,it has been widely used to estimate the density and its distribution function because of its simple form and excellent properties.Based on the Bernstein polynomial and its theory,we study the Bernstein polynomial model for estimating the density function based on grouped data and multivariate random variables.The coefficients of the Bernstein polynomial model can be obtained by an EM algorithm,and the optimal degree of the model can be estimated using a change-point method under the selected set M.In theory,the convergence rate of the Bernstein polynomial density estimator of grouped data is proved to be almost parametric.In the simulation,the sample size can choose as n=20,50,100,200,,bivariate normal,joint beta distribution are selected for simulation,and the mean integrated square error(MISE)of the multivariate Bernstein polynomial estimation and kernel estimation method can be calculated by R software,and the simulation results show that the Bernstein polynomial method is more efficient than the kernel density estimation.In the empirical analysis,the Bernstein polynomial method is used to estimate the density function of vehicle insurance claims provided by insuranceservice center,and compared with the kernel density estimation.The results show that the Bernstein polynomial estimation is more accurate.