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Generalised Exponential Kernels for Nonparametric Density Estimation

Laura M. Craig, Wagner Barreto-Souza

Abstract

This paper introduces a novel kernel density estimator (KDE) based on the generalised exponential (GE) distribution, designed specifically for positive continuous data. The proposed GE KDE offers a mathematically tractable form that avoids the use of special functions, for instance, distinguishing it from the widely used gamma KDE, which relies on the gamma function. Despite its simpler form, the GE KDE maintains similar flexibility and shape characteristics, aligning with distributions such as the gamma, which are known for their effectiveness in modelling positive data. We derive the asymptotic bias and variance of the proposed kernel density estimator, and formally demonstrate the order of magnitude of the remaining terms in these expressions. We also propose a second GE KDE, for which we are able to show that it achieves the optimal mean integrated squared error, something that is difficult to establish for the former. Through numerical experiments involving simulated and real data sets, we show that GE KDEs can be an important alternative and competitive to existing KDEs.

Generalised Exponential Kernels for Nonparametric Density Estimation

Abstract

This paper introduces a novel kernel density estimator (KDE) based on the generalised exponential (GE) distribution, designed specifically for positive continuous data. The proposed GE KDE offers a mathematically tractable form that avoids the use of special functions, for instance, distinguishing it from the widely used gamma KDE, which relies on the gamma function. Despite its simpler form, the GE KDE maintains similar flexibility and shape characteristics, aligning with distributions such as the gamma, which are known for their effectiveness in modelling positive data. We derive the asymptotic bias and variance of the proposed kernel density estimator, and formally demonstrate the order of magnitude of the remaining terms in these expressions. We also propose a second GE KDE, for which we are able to show that it achieves the optimal mean integrated squared error, something that is difficult to establish for the former. Through numerical experiments involving simulated and real data sets, we show that GE KDEs can be an important alternative and competitive to existing KDEs.
Paper Structure (8 sections, 3 theorems, 41 equations, 5 figures, 3 tables)

This paper contains 8 sections, 3 theorems, 41 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Generalised exponential kernels
  3. Monte Carlo experiments
  4. An alternative GE KDE with optimal MISE
  5. Real data applications
  6. Mexican Institute of Social Security data
  7. Snow data
  8. Concluding remarks

Key Result

Theorem 2.1

Assume that $f(\cdot)$ is a three-times differentiable function with bounded third derivative. Then, as $b\rightarrow0$, where $\gamma\approx0.577216$ is the Euler's constant.

Figures (5)

  • Figure 1: Boxplots of the MISEs based on the Configurations A, B, and C.
  • Figure 2: Boxplots of the MISEs based on the Configurations D and E.
  • Figure 3: Boxplots of the MISEs based on Configuration F.
  • Figure 4: Kernel density estimates and histogram for the Mexican Institute of Social Security data.
  • Figure 5: Kernel density estimates and histogram for the snow data.

Theorems & Definitions (8)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 4.1
  • proof
  • Remark 4.2
  • Remark 4.3