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Conditions for Unbiased Estimation Using Inverse Divergence


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.
Quotes
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Deeper Inquiries

How do the conditions for the function f differ between the IGT and GIGT mixture distributions, and what are the implications of these differences

The conditions for the function f differ between the Inverse Gaussian Type (IGT) and Generalized IGT (GIGT) mixture distributions in the context of the unbiased estimating equation. For the IGT distribution, the condition for the function f to satisfy the unbiased estimating equation without a bias correction term is given by the integral $\int_{0}^{\infty} \frac{g(t)f'(t)}{\sqrt{t} + 1} dt < \infty$. On the other hand, for the GIGT mixture distribution, the condition is $\int_{0}^{\infty} g(t)f'(t) dt < \infty$. These differences in conditions for the function f have implications on the types of functions that can be used for estimation in each distribution. The specific form of the function f determines the robustness of the estimator against outliers and influences the consistency of the estimator. Therefore, selecting the appropriate function f based on the distribution being used is crucial for achieving unbiased estimation without the need for bias correction terms.

What other types of Bregman divergences, beyond the inverse divergence, could potentially satisfy the unbiased estimating equation without a bias correction term, and under what statistical models

Beyond the inverse divergence, other types of Bregman divergences that could potentially satisfy the unbiased estimating equation without a bias correction term are the Mahalanobis distance and the Continuous Bregman distributions. The Mahalanobis distance corresponds to elliptical distributions, while Continuous Bregman distributions are defined by a generating function g and a strictly convex function φ. These Bregman divergences have been shown to eliminate the need for bias correction terms in the unbiased estimating equation under certain conditions. Under the statistical models of the Mahalanobis distance and Continuous Bregman distributions, specific functions f can be chosen to meet the conditions for unbiased estimation. By exploring different combinations of statistical models, Bregman divergences, and functions f, it is possible to identify scenarios where the unbiased estimating equation holds without requiring bias correction terms, enhancing the robustness and accuracy of parameter estimation.

What are the potential applications and practical implications of the unbiased estimating equation using inverse divergence, particularly in the context of robust parameter estimation

The unbiased estimating equation using inverse divergence has significant applications and practical implications in the context of robust parameter estimation. By satisfying the conditions for unbiased estimation without bias correction terms, the inverse divergence allows for robust estimation in the presence of outliers and heavy-tailed distributions. This method provides a framework for parameter estimation that is less sensitive to extreme data points and can yield more reliable estimates in challenging data scenarios. Practically, the unbiased estimating equation using inverse divergence can be applied in various fields such as finance, healthcare, and engineering where robust parameter estimation is crucial. It can help in modeling complex data distributions, identifying outliers, and improving the overall accuracy of statistical inference. By leveraging the properties of inverse divergence and selecting appropriate functions f, researchers and practitioners can enhance the robustness and reliability of their parameter estimation processes.
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