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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.

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