Probability Density Functions Estimation Based On Tight Wavelet Frames

In recent years,probability density function estimation has received more and more attention in nonparametric estimation.As an effective tool,wavelet has also been widely used in nonparametric probability density function estimation.The classical nonpara-metric density estimation problem is:(?,F,P)for the probability measure space,X is a continuous random variable,it obey the probability density function of f(x)un-known.How to go from n number independent identically distributed random sample X1,X2,…,Xn define an appropriate estimator fn so that it approximates f(x)in some sense.In statistics,there are two common methods to measure the merits of estimators.One is the mean square error(abbreviated as MSE)which examines the local error of the estimator and the real function.The other is a integral mean square error(shorthand for MISE)it estimator and true density function under the L2 meaning the overall error.In fact,in practical problems,the distribution of most data is difficult to assume in ad-vance.At the same time,considering that the reliability of estimation is to be improved as much as possible,it is a good choice to adopt a more adaptive density estimation method.Currently,wavelet methods are often based on orthogonal wavelet bases.Its advantage is that wavelet not only has good multi-resolution and local properties in time and frequency domain,but also can characterize a large class of function spaces.However,there is no correlation between the coefficients of the orthogonal wavelet basis expansion,which affects the accuracy of the estimate.In order to overcome this shortcoming,a tight wavelet frame with redundancy is used in this paper.This kind of framework in-herits the multi-scale structure of orthogonal wavelet bases and has correlation among decomposition coefficients,so it overcomes the shortcomings of orthogonal wavelet bases in essence.Firstly,the characterization of continuous Sobolev space based on compact wavelet framework is given.Secondly,an estimator of f(x)is fj1(x)is given based on the compact wavelet framework.The intrinsic relation between wavelet kernel and wavelet kernel is deduced.And then to L2(R)on continuous estimate the probability density function of a random variable.Finally,for the estimation of probability density function,we discuss the random error and deterministic error respectively,and an upper bound of their sum.On this basis,the kernel density estimation is used in the process of estimating the probability density function,so the expression of the kernel function in continuous form is deliberately given.Finally,the value of the parameter which can minimize the error is obtained.