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

Anwar Bataihah

doi:10.46481/jnsps.2026.3590

Authors

Anwar Bataihah anwerbataihah@gmail.com Department of Mathematics, Faculty of Science, Jadara University, Irbid, Jordan

Keywords:

Kernel density estimation, shifted exponential distribution, hazard rate, 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.

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