Wavelet Estimations Of Density Derivatives Based On Negatively Associated Sample

One of important applications of wavelets is nonparametric estimation. A perfectachievement has been made for wavelet density estimation by Donoho and etc, whenthe sample are independent and identically distributed (D. Donoho, I. Johnstone, G.Kerkyacharian, D. Picard. Density estimation by wavelet thresholding. Ann. stat.1996,24:508-539). Based on that work, Huiying Wang (in her doctoral thesis) studies waveletestimations for density derivatives by using the non-standard form of the diferentialoperations. It turn out that her results are optimal (sub-optimal) in some cases.In practical applications, a negatively associated sample is particularly important.Rao, Chaubey and etc have done a lot of work in this area, but their results are notgood enough. In this paper, we try to do something in that direction. A linear waveletestimation for density derivatives is given firstly, our result improves Chaubey’s theo-rem.(Y. P. Chaubey, H. Doosti, B. L. S. Prakasa Rao. Wavelet based estimation of thederivative of a density for a negatively associated process. Proceedings of The9th Is-lamic Countries Conference on Statistical Sciences,2007); Then motivated by Donoho’swork, we show a nonlinear wavelet estimation for a density. Finally, an estimation fordensity derivatives is proved. When the random sample are independent, our result isthe same as Liu and wang’s Theorem (Youming Liu, Huiying Wang. Wavelet estimationsfor density derivatives. Science China (Mathematics),2013,56(3):483-495).