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On the strong uniform consistency of a new kernel density estimator. (English) Zbl 0744.62052
The authors propose a multivariate kernel density estimator of the form $f_ n(x)=(n|\hbox{det} H_ n|)^{-1}\sum_{i=1}^ n K(H_ n^{-1}(x-\Phi_ n(X^*_ n,X_ i))),\qquad x\in\mathbb{R}^ k,$ where $$H_ n$$ is a nonsingular $$k\times k$$ matrix depending on the sample $$X^*_ n=(X_ 1,\dots,X_ n)$$ and $$\Phi_ n(\cdot,\cdot)$$ is a function from $$\mathbb{R}^{k\times n}\times\mathbb{R}^ k$$ to $$\mathbb{R}^ k$$. They prove its strong consistency (under certain regularity conditions). This result is then applied to a kernel whose mean vector and covariance matrix are $$\mu_ n$$ and $$V_ n$$, respectively, where $$\mu_ n$$ is an unspecified estimator of the mean vector of the density $$f$$ to be estimated and $$V_ n$$, up to a multiplicative constant, is an estimator of the covariance matrix of $$f$$.

MSC:
 62G07 Density estimation 62H12 Estimation in multivariate analysis 62G20 Asymptotic properties of nonparametric inference
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References:
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