Can smoothing an exponential distribution with different kernels create genuinely new probability models?
This study examines fixed-bandwidth kernel density estimation as a convolution operator applied to an exponential distribution.
The analysis considers compactly supported kernels, including the uniform, triangular, and Epanechnikov kernels, and investigates how their shape and bandwidth affect the expected kernel density estimate.
The main result shows that, away from the boundary and for values greater than or equal to the bandwidth, the expected estimate retains the original exponential shape. The kernel and bandwidth change only a multiplicative constant, not the functional form of the distribution.
After normalization over the interior region, the resulting density becomes exactly a shifted exponential distribution. Its hazard rate therefore remains constant and equal to the original exponential rate parameter.
This means that changing between compactly supported kernels does not create a genuinely new parametric family from an exponential baseline under fixed-bandwidth smoothing.
The study also clarifies the role of boundary effects. Near zero, the integration limit depends on the evaluation point, so the exact factorization breaks down and boundary bias appears.
A comparison with the Gaussian kernel shows why compact support matters. Because the Gaussian kernel extends across the entire real line, its expected estimate contains an additional position-dependent factor and does not preserve the exponential form exactly at finite values.
Numerical results support the theoretical findings: after normalization, the uniform, triangular, and Epanechnikov kernel estimates coincide with the shifted exponential curve, while the Gaussian result shows a small deviation.
The work provides a caution for researchers developing new statistical distributions. Classical fixed-bandwidth kernel smoothing of an exponential baseline does not, by itself, produce new hazard-rate behaviour or a new distributional family.
To obtain genuinely different models, the study suggests using variable bandwidths, non-translation-invariant kernels, nonlinear transformations, mixture constructions, or alternative baseline distributions such as Weibull, gamma, or lognormal models.
📖 Read the full article here:
https://doi.org/10.46481/jnsps.2026.3590
Published in: Journal of the Nigerian Society of Physical Sciences