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Empirical kernel transforms of parameter-estimated empirical processes. (English) Zbl 0586.62072
Let $$d\geq 1$$ be an integer and let $$X_ 1,X_ 2,..$$. be a sequence of independent d-dimensional vectors with common distribution function F(x), $$x\in R^ d$$. We assume that a parametric family of d-variate distribution functions is given ${\mathcal F}=\{F(x,\theta):\quad x\in R^ d;\quad \theta \in \Theta \subset R^ p\}$ and the common distribution of $$X_ 1,X_ 2,..$$. belongs to this family, i.e., there is a parameter $$\theta_ 0\in \Theta$$ so that $$F(x)=F(x;\theta_ 0)=F_ 0(x)$$. The true value of $$\theta_ 0$$ is unknown. Consider the estimated empirical process defined by $$\beta_ n(x)=n^{1/2}(F_ n(x)-F(x;\theta_ n))$$, where $$F_ n$$ is the empirical distribution function of $$X_ 1,...,X_ n$$ and $$\theta_ n$$ is some estimator of $$\epsilon_ 0$$. We find a sequence of kernels $$\{k^ N(x,y)\}$$ such that $\int_{R^ d}k^ N(x,y)\beta_ n(x)dx,\quad y\in I^ q,$ converges weakly to a q- dimensional Wiener process as n,N go to infinity.

##### MSC:
 62G30 Order statistics; empirical distribution functions 60F17 Functional limit theorems; invariance principles 62F10 Point estimation 62G05 Nonparametric estimation 62G10 Nonparametric hypothesis testing 60F05 Central limit and other weak theorems 62F03 Parametric hypothesis testing