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Likelihood Geometry of the Gumbel's Type-I Bivariate Exponential Distribution

Pooja Yadav, Tanuja Srivastava

TL;DR

This paper investigates the maximum likelihood degree (ML-degree) of the association parameter $\theta$ in Gumbel's Type-I bivariate exponential distribution (GBED-I) using algebraic statistics. By rewriting the score equation as a rational function $\frac{f(\theta)}{g(\theta)}=0$ with $f$ and $g$ of degree $2n$, the authors analyze the geometry and multiplicities of common zeros to determine the ML-degree, which depends on the data. They derive explicit formulas for $m_d(\theta)$ under various configurations of shared zeros and double zeros among the $g_i(\theta)$, establishing bounds (up to $2n$) and illustrating with $n=3$ examples. The results provide a data-dependent, algebraic-geometry–based understanding of the estimator's complexity and guide computational approaches, while noting that closed-form MLEs are not available and real solutions may lie outside the natural parameter range.

Abstract

In algebraic statistics, the maximum likelihood degree of a statistical model refers to the number of solutions (counted with multiplicity) of the score equations over the complex field. In this paper, the maximum likelihood degree of the association parameter of Gumbel's Type-I bivariate exponential distribution is investigated using algebraic techniques.

Likelihood Geometry of the Gumbel's Type-I Bivariate Exponential Distribution

TL;DR

This paper investigates the maximum likelihood degree (ML-degree) of the association parameter in Gumbel's Type-I bivariate exponential distribution (GBED-I) using algebraic statistics. By rewriting the score equation as a rational function with and of degree , the authors analyze the geometry and multiplicities of common zeros to determine the ML-degree, which depends on the data. They derive explicit formulas for under various configurations of shared zeros and double zeros among the , establishing bounds (up to ) and illustrating with examples. The results provide a data-dependent, algebraic-geometry–based understanding of the estimator's complexity and guide computational approaches, while noting that closed-form MLEs are not available and real solutions may lie outside the natural parameter range.

Abstract

In algebraic statistics, the maximum likelihood degree of a statistical model refers to the number of solutions (counted with multiplicity) of the score equations over the complex field. In this paper, the maximum likelihood degree of the association parameter of Gumbel's Type-I bivariate exponential distribution is investigated using algebraic techniques.

Paper Structure

This paper contains 6 sections, 13 theorems, 65 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Geometry of the score equation
  4. The multiplicity of common zeros of and
  5. The ML degree of the association parameter
  6. Conclusion

Key Result

Corollary 3.1

If all $g_{i}(\theta)$ have distinct zeros, then $f(\theta)$ and $g(\theta)$ will have common zeros if and only if there exists some $k$ such that $f_{k}(\theta)$ and $g_{k}(\theta)$ have common zero.

Figures (3)

  • Figure 1: The likelihood function of the parameter $\theta$ given the sample $X_{1},X_{2},X_{3}$.
  • Figure 2: The likelihood function of the parameter $\theta$ given the sample $X_{1},X_{2},X_{3}$.
  • Figure 3: The likelihood function of the parameter $\theta$ given the sample $X_{1},X_{2},X_{3}$.

Theorems & Definitions (31)

  • Definition 2.1: Maximum likelihood degree
  • proof
  • Corollary 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 21 more