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| Convergence Rate of Multivariate K-Nearest Neighbor Density Estimates |
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Citation： |
Yang Zhenai.Convergence Rate of Multivariate K-Nearest Neighbor Density Estimates[J].Chinese Annals of Mathematics B,1990,11(4):536~545 |
| Page view： 932
Net amount： 791 |
Authors： |
Yang Zhenai; |
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| Abstract： |
Let \mathscr{F} be the collection of m-times continuously differentiable probability densities f on R^d such that |D^\alpha f(x_1)-D^\alpha f(x_2)|\leq M||x_1-x_2||^\beta for x_1,x_2\in R^d, [\alpha]=m, where D^\alpha denotes the differential operator defined by D^\alpha=\partial^[\alpha]/\partial x_1^\alpha_i \cdots \partial x_d^\alpha_d.Under rather weak conditions on K(x), the necessary and sufficient conditions for sup |f_n(x)-f(x)|=O((logn/n)^{\lambda /(d+3\lambda),\lambda=m+\beta,f\in \mathscr{F} are that \int\limits {x^\alpha K(xi)dx=0 for 0<[\alpha]\leq m.Finally the convergence rate at a point is given. |
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