L_1核密度估计的弱相合和完全相合性

设x_n的密度函数为f(x),X∈R~d,f_n(x)=(nh~d)~(-1)sum from i-1 to n k((x-x_i)/h)为f的核估计,其中0

The Weak and Complete Consistency in L_1 for Kernel Density Estimate.

Let f be a density on R~d and f_n be the kernel estimate of f, f_n (x) = (nh~d)~(-1) sum from i=1 to n K ((x-x_i)/h), where h=h_n→0 and K is an absolutely integrable function with ∫k(x)dx=1. Let J_n= f|f_n(x)-f(x)|dxwe give some sufficient conditions for J_n→0 in probability or J_n→0 complete- ly, where {x_n) is α-mixing or φ-mixing sample, which generalize the result in.