On estimation of a density and its derivatives期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

On estimation of a density and its derivatives

Authors:	K. F. Cheng

Affiliation:	(1) State University of New York at Buffalo, Buffalo, USA

Abstract:	Summary Letf _n ^(p) be a recursive kernel estimate off ^(p) thepth order derivative of the probability density functionf, based on a random sample of sizen. In this paper, we provide bounds for the moments of $left\| {f_n^{(p)} - f^{(p)} } right\|_{L_2 } = left[ {smallint [f_n^{(p)} (x) - f^{(p)} (x)]^2 dx} right]^{{1 mathord{left/ {vphantom {1 2}} right. kern-nulldelimiterspace} 2}}$ and show that the rate of almost sure convergence of $left\| {f_n^{(p)} - f^{(p)} } right\|_{L_2 }$ to zero isO(n ^−α), α<(r−p)/(2r+1), iff ^(r),r>p≧0, is a continuousL ₂(−∞, ∞) function. Similar rate-factor is also obtained for the almost sure convergence of $left\| {f_n^{(p)} - f^{(p)} } right\|_infty = mathop {sup }limits_x left\| {f_n^{(p)} (x) - f^{(p)} (x)} right\|$ to zero under different conditions onf. This work was supported in part by the Research Foundation of SUNY.

Keywords:	Primary 62G05 Secondary 60F15
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