异方差回归模型近邻型估计的相合性

CONSISTENCY OF A NEAREST NEIGHBOR ESTIMATION IN HETEROSCEDASTICITY REGRESSION MODEL

本文研究异方差回归模型Ｙ（ｎ）ｉ＝ｇ（ｘ（ｎ）ｉ）＋ε（ｎ）ｉ，ｉ＝１，…，ｎ，其中ｇ是未知实函数，ｘ（ｎ）ｉ是非随机设计点列，ε（ｎ）ｉ是随机误差．文中定义了一类ｇ（ｘ）的近邻型估计ｇｎ（ｘ）＝ｎｉ＝１Ｗｎｉ（ｘ）Ｙ（ｎ）ｉ，得到了ｒ阶平均相合和渐近正态性．特别，在∞ｎ＝１ｎｉ＝１Ｅ｜ε（ｎ）ｉ｜ｓ／（ｎｉ）ｓ／ｒ＜∞，ｍａｘ１≤ｉ≤ｎＥ｜ε（ｎ）ｉ｜ｓ＝Ｏ（ｎｓ／ｒ）（某ｓ＞ｒ＞２）下，获得了ｇｎ（ｘ）的强相合和一致强相合性．

In this paper, the heteroscedasticity regression model Y (n) i=g(x (n) i)+ε (n) i,i=1,…,n , is studied, where g is an unknown function, x (n) i are known and nonrandom,and ε (n) i are independent random errors. Under suitable conditions, a nearest neighbor estimate g n(x)=ni=1W ni (x)Y (n) i of g(x) is defined.It is shown that the g n is r th mean consistent and asymptotically normal.Especially,the strong consistency and uniform strong consistency for g n(x) are also obtained,provided that ∞n=1ni=1E|ε (n) i| s/(ni) s/r <∞ and max 1≤ i ≤ nE|ε (n) i| s =O(n s/r ) for some s>r >2.