On the effect of fixed-bandwidth kernel density estimation on the exponential distribution

Abstract

Kernel density estimation (KDE) is widely used as a nonparametric smoothing operator in statistics. In this work, we study fixed-bandwidth KDE as a convolution operator applied to an exponential baseline distribution with rate parameter (beta > 0). We show that, for any compactly supported kernel K on [-1,1] and any fixed bandwidth h > 0, the expected KDE admits an exact factorization in the interior region y  ge  h:  mathbb{E}[widehat f(y)] = beta e^{-beta y} C₀, where C₀ = (int_{-1}^{1}K(u)e^{beta h u} du) depends only on the kernel and bandwidth. Thus, the exponential density is an eigenfunction of the kernel-smoothing operator in the interior domain: its shape is preserved up to multiplication by the eigenvalue C₀. After normalization on (h, infty), the resulting distribution reduces exactly to the shifted exponential density (beta e^{-beta(y-h)}). The result clarifies the role of boundary effects in kernel smoothing and shows that fixed-bandwidth KDE does not generate new parametric families from exponential baselines under tail normalization.