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Weak convergence of $$k$$-NN density and regression estimators with varying $$k$$ and applications. (English) Zbl 0643.62027
Let $$(X_ i,Z_ i)$$, $$i\geq 1$$, be independent, two-dimensional random vectors distributed as (X,Z), where X has marginal distribution function F with density function f and where $$\mu (x)=E(Z| X=x)$$ is the regression function of Z at $$X=x$$. For fixed x, set $$Y_ i=| X_ i- x|$$, let $$Y_{n1}\leq...\leq Y_{nn}$$ denote the order statistics of $$Y_ 1,...,Y_ n$$, and let $$Z_{n1},...,Z_{nn}$$ be the induced order statistics in $$(Y_ 1,Z_ 1),...,(Y_ n,Z_ n)$$, i.e., $$Z_{ni}=Z_ j$$ if $$Y_{ni}=Y_ j.$$
The k-nearest neighbor (k-NN) estimator of f(x) corresponding to the uniform kernel, i.e., $$f_{nk}(x)=(k-1)/(2nY_{nk})$$, and the k-NN estimator of $$\mu$$ (x) with uniform weights, i.e., $$\mu_{nk}(x)=k^{- 1}\sum^{k}_{j=1}Z_{nj}$$, for fixed x and k varying in an appropriate range, are transformed into continuous time stochastic processes by setting $T_ n(t)=f_{n,[n^{4/5}t]}(x),\quad S_ n(t)=\mu_{n,[n^{4/5}t]}(x),\quad 0<a\leq t\leq b<\infty.$ Under the usual second-order smoothness conditions, it is shown that the two processes $\{n^{2/5}[T_ n(t)-f(x)],\quad a\leq t\leq b\},\quad \{n^{2/5}[S_ n(t)-\mu (x)],\quad a\leq t\leq b\}$ have a common limiting structure as the sample size n tends to infinity. These results lead to asymptotic linear models in which BLUE’s and suitably biased linear combinations of k-NN estimators with varying k are considered.
Reviewer: E.Häusler

##### MSC:
 62G05 Nonparametric estimation 62J02 General nonlinear regression 60F17 Functional limit theorems; invariance principles 62G30 Order statistics; empirical distribution functions
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