Bernstein Type Polynomial Estimation Of Quantile Function And Its Properties

In statistics,the quantile is a very important distribution feature.It has a high application background,such as critical values of hypothesis testing,endpoints of interval estimation,and is particularly important in the risk management of the financial industry,Va R and CVa R have a direct relationship with quantile,so it is of great significance to study quantile estimates.Common quantile estimation methods include empirical distribution quantile estimation and kernel quantile estimation.However,the former does not consider function smoothness,while the latter has some drawbacks such as boundary effects,slow convergence rate,and so on.Bernstein polynomials have been used for approximate estimation of functions in many fields because of their superior properties.The Bernstein estimation of continuous functions is very similar to the original function.Based on the Bernstein cumulative distribution function,this paper deduces the quantile estimation formula.The EM algorithm and the change-point detection method are used to find the maximum likelihood estimate ofmp and model optimal degree respectively.At present,there are few studies on the properties of Bernstein type polynomial estimation function.In this paper,we introduce the Bernstein type polynomial to estimate the density function property lemma,and prove that Bernstein type polynomial estimation quantiles have significant advantages in terms of mean square consistency and asymptotic normality,and the convergence rate of this estimation is obtained.In the simulation and empirical analysis,the normal distribution,uniform distribution were used to test the accuracy of the quantile estimation of the Bernstein polynomial at a given probability level,and compared with the kernel quantile estimation and empirical distribution quantile estimation.The simulation results show that Bernstein type polynomial estimation is significantly better than the other two methods,and is robust and efficient.In the empirical analysis,the daily logarithmic return rate of the above Shanghai Stock Index and Shenzhen Stock Index is the study object.The GARCH model and the Bernstein polynomials model,were used to estimate the Va R values of the two indices at different probability levels and compare their respective Va R values.The empirical results show that the Va R estimation of the Shanghai index are smaller than of the Shenzhen Component Index,and the risk ofinvesting in the Shenzhen Component Index is greater,while the Shanghai Composite Index is insured.This will help investors' investment decisions in the turmoil of the stock market.