The kernel estimate is relatively stable (Q1092554)

Consider the Parzen-Rosenblatt kernel estimator \(f_ n=(1/n)\sum ^{n}_{i=1}K_ h(x-X_ i)\), where \(h>0\) is a constant, K is an absolutely integrable function with integral one, \(K_ h(x)=(1/h^ d)K(x/h)\), and \(X_ 1,...,X_ n\) are i.i.d. random variables with common density f on \(R^ d\). We show that for all \(\epsilon >0\), \[ \sup _{h>0,f}P(| \int | f_ n-f| -E\int | f_ n-f| | >\epsilon)\leq 2 \exp (-n\epsilon ^ 2/32\int | K|). \] We also establish that \(f_ n\) is relatively stable, i.e. \[ \int | f_ n-f| /E\int | f_ n-f| \to 1\quad in\quad probability\quad as\quad n\to \infty, \] whenever lim inf \(\sqrt{n}E\int | f_ n-f| =\infty\). We also study what happens when h is allowed to depend upon the data sequence.