正交级数密度估计积分平方误差的极限定理_英文_

正交级数密度估计积分平方误差的极限定理_英文_ JOURNAL OF CHINA UNIVERSITY OF SCIENCE AND TECHNOLO GY 2 0 0 1 年 8 月 Aug. 2 0 0 1 () Article ID :025322778 20010420407205 L imit Theorem f or Integrated Square Error Ξof Orthogonal Series Density Estimators WU Cheng2qing , ZHAO Lin2cheng ( )Depantment of Statisties and Finance , US TC , Hefei , Anhui 230026 ( ) Abstract : The limiting distribution for is derived integrated square error ISEof nonparamet2 ric density estimators based on orthogonal series under some mild conditions , and it is proved that the limiting distribution of the standardization of ISE is the distribution of a linear combi2 2 χnation of independent centred random variables. 1 Key words :central limittheorem ; orthogonal series ; density estimate ; integrated square error CL C number :O212 . 7 Document code :A AMS Subject Cla ssif ications( 2000) : Primary 62 E20 ,62 H11 ; Secondary 62 G05 1 Introduction d Let X be a random variable taking values in Rwith a square2integrable density f . It is well known that the density can be written as ? α( ) ( ) f x= 1/ 2ni 5 d () and d > 1 and the shrinkage sequences considered by Hallwith b= max 1 - ci , 0for c ? ni - γ(γ ) const . n > 1/ 2and d > 0. 2 L imiting Distribution f or ISE ) ( ) ( ) ( For simplicity we write ^f x bas f x. Notice that {