Core Concepts
The conditions under which the unbiased estimating equation holds without a bias correction term for loss functions composed of a monotonically increasing function f and inverse divergence are characterized for two types of statistical models: the inverse Gaussian type (IGT) distribution and the generalized inverse Gaussian type (GIGT) mixture distribution.
Abstract
This paper focuses on the Bregman divergence defined by the reciprocal function, called the inverse divergence. It clarifies the combinations of the statistical model and function f that eliminate the bias correction term when using inverse divergence for estimation.
For the IGT distribution, the unbiased estimating equation holds without a bias correction term if and only if the integral of the product of the generating function g and the derivative of the function f is bounded. This condition is different from the condition for the GIGT mixture distribution, which is a generalization of the continuous Bregman distribution.
The paper also extends the results to the multi-dimensional case by expressing the inverse divergence as a linear sum over the dimensions. The corresponding statistical model, the multivariate IGT (MIGT) distribution, is newly defined, and the condition for the function f is provided as a double integral.
The IGT and GIGT mixture distributions are special cases of the regular exponential family, and the inverse divergence is a special case of the β-divergence. This suggests that the inverse divergence is a unique Bregman divergence that ensures the unbiased estimating equation without a bias correction term for certain statistical models.
Stats
The expected value of the IGT distribution always exists, independent of the generating function g, and satisfies E[X] = θ.
The expected value of the MIGT distribution exists, independent of the generating function g, and satisfies E[X] = θ.