关于核密度估计误差准则的渐近性质

Asmptotic Properties of Error Criterion for Kernel Density Estimate

假定f^d(、)是R上的一个密度函数,(?)(·;h)是这个密度函数建基于来自这个密度函数f(·)的容量为n的一个随机样本的核型估计.令ASE(h),ISE(h)和MISE(h)分别表示这个非参数密度估计量的平均均方误差、积分均方误差以及平均积分均方误差.在某些一般的条件下,我们证明了ASE(h)/MISE(h)以及ISE(h)/MISE(h)

Suppose that f( · ) is a density function on Rd. f ( ·; h) are kernel type estimators of the density function f ( · )based on a random sample of size n from the density function f( · ). Let ASE (h)、 ISE (h) and MISE (h) denote averaged squared error. integral square error and averaged integval square error of the nonparametric density eetimators respectively. We prove the almost sure conrergence of ASE (h) /MISE (h) , to one, and of ISE(h)/ MISE(h), to one, under certain general conditions.