On the strong uniform consistency of a new kernel density estimator (Q1813631)

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 \(f\).