Convergence in Distribution of Self-Normalized Sup-Norms of Kernel Density Estimators

Abstract

Let f_n denote a kernel density estimator of a density f on the real line, for a bounded, compactly supported probability kernel. Under relatively weak smoothness conditions on f and K it is proved, for every 0 < β < 1/2, that the sequence

$${{\hat{A}}_{n}}\left( {\frac{{\sqrt {{n{{h}_{n}}}} }}{{\parallel K{{\parallel }_{2}}\parallel {{f}_{n}}\parallel _{\infty }^{{1/2 - \beta }}}}\mathop{{\sup }}\limits_{{t \in {{{\hat{D}}}_{{{{a}_{n}}}}}}} \frac{{|{{f}_{n}}(t) - f(t)|}}{{f_{n}^{\beta }(t)}} - {{{\hat{A}}}_{n}}} \right)$$

converges in distribution to the double exponential law. Here ${{\hat{A}}_{n}}$ is constructed from the sample, a _n → ∞ as a power of n and ${{\hat{D}}_{{{{a}_{n}}}}} = \{ t:{{f}_{n}}(t) \geqslant a_{n}^{{ - 1}}\}$. Thus, this result provides distribution free asymptotic confidence bands for densities on the real line.