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On the shapes of bilateral Gamma densities

Uwe Küchler, Stefan Tappe

TL;DR

The paper delivers a comprehensive analysis of bilateral Gamma densities, addressing how to represent, smooth, and characterize their shapes for fitting real data. It derives density representations via Whittaker and related special functions, establishes smoothness and unimodality through self-decomposability, and details asymptotic behavior at zero and in the tails. By classifying density shapes across parameter regimes and connecting to the Variance Gamma family, it provides practical guidance for parameter inference and model selection in applications with heavy tails and asymmetry, notably in finance. These results enable accurate density plotting and fitting for bilateral Gamma models across diverse datasets.

Abstract

We investigate the four parameter family of bilateral Gamma distributions. The goal of this paper is to provide a thorough treatment of the shapes of their densities, which is of importance for assessing their fitting properties to sets of real data. This includes appropriate representations of the densities, analyzing their smoothness, unimodality and asymptotic behaviour.

On the shapes of bilateral Gamma densities

TL;DR

The paper delivers a comprehensive analysis of bilateral Gamma densities, addressing how to represent, smooth, and characterize their shapes for fitting real data. It derives density representations via Whittaker and related special functions, establishes smoothness and unimodality through self-decomposability, and details asymptotic behavior at zero and in the tails. By classifying density shapes across parameter regimes and connecting to the Variance Gamma family, it provides practical guidance for parameter inference and model selection in applications with heavy tails and asymmetry, notably in finance. These results enable accurate density plotting and fitting for bilateral Gamma models across diverse datasets.

Abstract

We investigate the four parameter family of bilateral Gamma distributions. The goal of this paper is to provide a thorough treatment of the shapes of their densities, which is of importance for assessing their fitting properties to sets of real data. This includes appropriate representations of the densities, analyzing their smoothness, unimodality and asymptotic behaviour.

Paper Structure

This paper contains 7 sections, 6 theorems, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Bilateral Gamma distributions
  3. Representations of the densities
  4. Smoothness
  5. Unimodality
  6. Asymptotic behaviour
  7. Shapes of the densities

Key Result

Proposition 3.1

Assume $\alpha^+ \in \mathbb{N} = \{ 1,2,\ldots \}$. Then it holds for each $x \in (0,\infty)$ where the coefficients $(a_k)_{k=0,\ldots,\alpha^+ - 1}$ are given by

Figures (1)

  • Figure 1: The shapes of $f$ for $\lambda^+ = \lambda^-$. Different choices of $\lambda^+$ and $\lambda^-$ may shift the mode and change the skewness.

Theorems & Definitions (12)

  • Proposition 3.1
  • proof
  • Theorem 4.1
  • proof
  • Proposition 4.2
  • proof
  • Theorem 5.1
  • proof
  • Proposition 5.2
  • proof
  • ...and 2 more