Computational algebraic methods in efficient estimation

TL;DR

This paper explores the use of algebraic geometry and computational algebraic methods to construct efficient estimators in statistical models, linking information geometry with algebraic statistics and demonstrating practical computational techniques.

Contribution

It introduces algebraic methods for constructing first and second order efficient estimators and applies Gr"obner basis techniques to simplify estimating equations.

Findings

Algebraic estimators can be constructed using purely algebraic equations.

Gr"obner basis methods reduce the complexity of estimating equations.

Feasible computational approaches like homotopy continuation are applicable for solving polynomial equations.

Abstract

A strong link between information geometry and algebraic statistics is made by investigating statistical manifolds which are algebraic varieties. In particular it it shown how first and second order efficient estimators can be constructed, such as bias corrected Maximum Likelihood and more general estimators, and for which the estimating equations are purely algebraic. In addition it is shown how Gr\"obner basis technology, which is at the heart of algebraic statistics, can be used to reduce the degrees of the terms in the estimating equations. This points the way to the feasible use, to find the estimators, of special methods for solving polynomial equations, such as homotopy continuation methods. Simple examples are given showing both equations and computations. *** The proof of Theorem 2 was corrected by the latest version. Some minor errors were also corrected.