Unbiased Estimating Equation on Inverse Divergence and Its Conditions

TL;DR

This paper investigates conditions for unbiased estimating equations based on inverse divergence, characterizing specific statistical models and extending the framework to multi-dimensional cases.

Contribution

It clarifies conditions for unbiasedness of estimating equations using inverse divergence for certain models and extends the analysis to multi-dimensional settings.

Findings

Characterized conditions for unbiased estimating equations with inverse divergence.

Identified specific models: inverse Gaussian and generalized inverse Gaussian mixtures.

Extended divergence analysis to multi-dimensional cases.

Abstract

This paper focuses on the Bregman divergence defined by the reciprocal function, called the inverse divergence. For the loss function defined by the monotonically increasing function $f$ and inverse divergence, the conditions for the statistical model and function $f$ under which the estimating equation is unbiased are clarified. Specifically, we characterize two types of statistical models, an inverse Gaussian type and a mixture of generalized inverse Gaussian type distributions, to show that the conditions for the function $f$ are different for each model. We also define Bregman divergence as a linear sum over the dimensions of the inverse divergence and extend the results to the multi-dimensional case.