Density Estimation Based On Wavelet First-Order Sobol Indices And Multiwavelet Biased Data

In statistics,the problem of density estimation has always been one of the core research areas in non-parametric estimation.In addition to traditional histogram and kernel estimation methods,wavelet density estimation is one of the research hotspots in recent years.This is mainly due to excellent characteristics of wavelet base such as orthogonality,compact support,and multiresolution analysis(MRA).Density estimation has a good application in many aspects,such as:global sensitivity analysis(GSA).The most common sensitivity method is the sobol index,variance is used to decompose the model into a function of a single parameter and its mutual combination;the importance of the parameter in the model is measured by the ratio of the variance to the total output.In the sampling process,the data is basically noisy data,and the obtained density function has a large error.It is necessary to study the density estimation with the noise model.The noise model due to the uneven sampling opportunity is called the biased model(biased data).In the first part of this paper,the wavelet density estimation method is applied to the calculation of sobol index,and the wavelet nonparametric estimation expression of sobol index is given and the upper bound of convergence is proved to be O(n-(2a-3)/(2a+1)in the sense of mean square error(MSE).The numerical simulation experiment gives a comparison between the new method and the Monte Carlo method.Since the wavelet basis function can not satisfy the orthogonality,symmetry and tight support at the same time,the multiwavelet can satisfy the three properties of appeal at the same time.In the second part,we present our method for multiwavelet estimation for a density function based on random samples from a weighted distribution,we propose linear and nonlinear multiwavelet estimators for the biased data,we prove the upper bound in the sense of integral mean error(MISE).Finally,we provide simulation and real data experiments and comparison with wavelet method.to illustrate the effectiveness and behavior of the proposed method and estimator.