The Jacobi Theta Distribution

TL;DR

The paper introduces the Jacobi theta distribution, a new continuous probability distribution derived from exponential variables, with unique properties and applications in modeling, expressed via Jacobi theta functions.

Contribution

It defines and characterizes the Jacobi theta distribution, including its properties, approximations, inference methods, and potential applications in modeling.

Findings

Supported on positive reals with a single parameter

Expressed using Jacobi theta functions for density and cumulative functions

Includes asymptotic and log-normal approximations

Abstract

We form the Jacobi theta distribution through discrete integration of exponential random variables over an infinite inverse square law surface. It is continuous, supported on the positive reals, has a single positive parameter, is unimodal, positively skewed, and leptokurtic. Its cumulative distribution and density functions are expressed in terms of the Jacobi theta function. We describe asymptotic and log-normal approximations, inference, and a few applications of such distributions to modeling.