CONVERGENCE RATE OF ESTIMATES OF PROBABILITY DENSITY FUNCTION AND ITS DERIVATIVES

In this paper, we obtain the strong uniform convergence rate δ_n（r） of estimatis of probability density f(x) and its derivatives f^（r）（x）, here \begingathered\delta_n(r)=\sup _-\infty<\infty<\infty\left|\hatf_n^(r)(x)-f^(r)(x)\right|, \hatf_n^(r)(x)=\frac1n \pi \sum_k=1^n\left\frac\sin \left\left(x-\infty_k\right) \varphi(n)\rightx-x_k\right_0^(r), \\ (r=0,1,2, \cdots),\endgathered, x_k（k=1, 2,…, n） are i.i.d. samples drawn from a population with density f（x）, \varphi(n)>0, \lim _i \rightarrow \infty \varphi(n)=\infty,The asymptotic normalities are discussed.