The Maximum Likelihood Degree of Farlie Gumbel Morgenstern Bivariate Exponential Distribution
Pooja Yadav, Tanuja Srivastava
TL;DR
The paper investigates the maximum likelihood degree (ML-degree) for the association parameter $\theta$ of the Farlie–Gumbel–Morgenstern (FGM) bivariate exponential distribution. It reformulates the score equation into a rational form by introducing $c_i$ via $\frac{1}{c_i}=(2 e^{-x_i}-1)(2 e^{-y_i}-1)$, yielding $\sum_{i=1}^n \frac{1}{\theta+c_i}=0$, and analyzes the zeros of the numerator $h(\theta)$ relative to the denominator $k(\theta)$. The main result provides a closed-form ML-degree: if the distinct $c_i$'s have multiplicities $n_i$ with $l$ values repeated (and $m=\sum_{i=1}^l n_i$), then ML-degree $= n+l-m-1$, with the special case of all $c_i$ distinct giving ML-degree $n-1$. This work yields an explicit algebraic criterion for the number of complex critical points of the likelihood in this model, clarifying how data-induced degeneracy affects inference in reliability, queueing, and actuarial applications.
Abstract
The maximum likelihood degree of a statistical model refers to the number of solutions, where the derivative of the log-likelihood function is zero, over the complex field. This paper examines the maximum likelihood degree of the parameter in Farlie-Gumbel-Morgenstern bivariate exponential distribution.