Estimation Of A Mixed Bernstein Polynomial Model With Multidimensional Variable Density

On the basis of density function estimation method,in this paper,we main-ly study the problem of estimating density functions by using multidimensional Bernstein polynomials defined on simplex.Different from previous studies on mul-tidimensional density estimation,the multidimensional density estimation studied in this paper is based on the simplex region,and there are certain constraints between variables.Inspired by the Bernstein polynomial estimation model in one-dimensional case,the mixed model of multidimensional Bernstein polynomials can be used to estimate the unknown multidimensional density,and the result is very good.This paper mainly focuses on two-dimensional situation.The EM algorithm is used to calculate the optimal two-dimensional Bernstein mixed model in the sense of maximum likelihood.Using the samples collected under the density function to fit,it can be seen that when the order m increases,the change of maximum likeli-hood function value of the hybrid model decreases gradually.At the same time,it can be seen that the mean square error between the estimated value and the real da-ta decreases first and then increases.Then the optimal order of the two-dimensional Bernstein polynomial mixed model can be found by the method of change point.Combining with fitting simulation,we can see that the fitting effect of the model is good.Based on the two-dimensional Bernstein polynomial asymptotic expansion on simplex,the asymptotic effect of the mixed model is proved.On the basis of the law of large numbers and the central limit theorem,the convergence of the mixed model under large samples is illustrated.