密度估计函数的收敛速度及重对数率

设ｆＦ为（－∞，∞）上的一族概率密度，ｘ１，ｘ２，…，ｘｎ为取自ｆ的样本。记Ｊｎｉ＝（（ｉ－１）ｈｎ，ｉｈｎ），ｈｎ∞（ｎ→∞），又记Ｒｉ＝＃｛ｔ：ｔ＝１，２，…，ｎ｝，当ｘＪｎｉ时，讨论了ｆ（ｘ）的密度估计函数。并且在Ｌｉｐｓｈｉｔｚ条件下研究了密度估计函数ｆｎ（ｘ）的渐近正态性，最佳可能收敛速度和一致收敛的重对数率。当０＜α＜１，β＜１－α２时，ｆｎ（ｘ）－ｆ（ｘ）＝Ｏ（ｌｎｎｎ－β）ａ．ｓ．；当－１４＜α＜１２时，ｓｕｐｘ｜ｆｎ（ｘ）－ｆ（ｘ）｜＝Ｏ（ｎα－１２ｌｎｌｎｎ）ａ．ｓ等．

Convergence Rate of Some Density Estimation Function and Its Law of the Iterated Logarithm

Let fF be a family propabitily density on (-∞,∞).x 1,x 2,…,x n be a random sample from f.Let J ni =((i-1)h n,ih n),h n→∞(n→∞),R i=#{t:t=1,2,…,n,x t  J ni }.When xJ ni ,f n(x)=R inh n for estimators of functionals of density was discussed. And its character of asymtotially normal distribution,best possible convergence rate and the law of the iterated logarithm of uniformly convergence were given. Such as when 0<α<1,h n=n -α ,β<1-α2,f n(x)-f(x)=O(ln n n -β ) a.s.; When-14<α<12,supx|f n(x)-f(x)|=O(n α-12 ln ln n)a.s. and so on.