φ混合样本回归函数核估计的强一致收敛速度

THE STRONG UNIFORM CONVERGENCE RATES OF KERNEL ESTIMATES OF REGRESSION WHEN SAMPLE IS A φMIXING SEQUENCE

设Ｘ和Ｙ分别是ｄ维和１维随机变量，（Ｘ，Ｙ）～Ｆ（ｘ，ｙ）。（Ｘｊ，Ｙｊ），ｊ＝１，２，…，ｎ为来自（Ｘ，Ｙ）的样本。讨论了当样本为平稳φ混合随机序列时，回归函数ｍ（ｘ）＝Ｅ（Ｙ｜Ｘ＝ｘ）的核估计ｍｎ（ｘ）（Ｎａｄａｒａｙａ于１９６４年提出的）的强一致收敛速度。在其他条件不变的情况下，得出了与独立样本相同时的强一致收敛速度。

Assuming that random vector (X,Y)∈Rd×R1 and (X,Y)～F(x,y),the strong uniform convergence rates of kernal estimate mn(x) which was proposed by Nadaraya in 1964 for regression function m(x)=E(Y|X=x) is studied when sample (Xj,Yj)∈Rd×R1,j=1,2,… is a stationary φmixing sequence.The same strong uniform convergence rates as those when sample is an i.i.d. sequence are obtained with the other assumptions unchanged.