Unbiased Estimating Equation on Inverse Divergence and Its Conditions
This work provides theoretical conditions for unbiased estimation in specific statistical models, which is incremental for researchers in statistical inference and divergence-based methods.
The paper tackles the problem of establishing unbiased estimating equations for Bregman divergence based on the reciprocal function, characterizing conditions for statistical models and functions to achieve this, specifically for inverse Gaussian and mixture of generalized inverse Gaussian distributions, and extends the results to multi-dimensional cases.
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.