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

Let fn 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 Ân ( √ nhn ‖K‖2‖fn‖ ∞ sup t∈D̂an |fn(t) − f(t)| f n (t) − Ân ) converges in distribution to the double exponential law. Here Ân is constructed from the sample, an → ∞ as a power of n and D̂an = {t : fn(t) ≥ a−1 n }. Thus, this result provides distribution free asymptotic confidence bands for densities on the real line.