Density estimation for samples satisfying a certain absolute regularity condition
Let \(\{\xi_ i\}\) be a strictly stationary sequence of random variables which are defined on a probability space (\(\Omega\),\({\mathcal A},P)\). For \(a\leq b\), let \({\mathcal M}^ b_ a\) denote the \(\sigma\)-algebra of events generated by \(\xi_ a,...,\xi_ b\). The sequence \(\{\xi_ i\}\) satisfies the absolute regularity condition if \[ \beta (n)=E\{\sup_{A\in {\mathcal M}_ n^{\infty}}| P(A| {\mathcal M}^ 0_{-\infty})-P(A)| \}\downarrow 0\quad (n\to \infty). \] Let f(x) be the density function of \(\xi_ 1\). As an estimate of f(x) the author considers \(f_ n(x)=n^{-1}\sum^{n}_{j=1}H_ n(x,\xi_ j)\), where \(H_ n(x,y)\) is a Borel-measurable function defined on a Euclidean space. In this paper asymptotic properties of the probability \(P(\sup_ x| f_ n(x)-f(x)| >\epsilon)\) are studied.
scientific article; zbMATH DE number 3878160