Asymptotic expansion of the risk of maximum likelihood estimator with respect to α-divergence

ABSTRACT

For a given parametric probability model, we consider the risk of the maximum likelihood estimator with respect to α-divergence, which includes the special cases of Kullback–Leibler divergence, the Hellinger distance, and essentially χ²-divergence. The asymptotic expansion of the risk is given with respect to sample sizes up to order n^{− 2}. Each term in the expansion is expressed with the geometrical properties of the Riemannian manifold formed by the parametric probability model.

KEYWORDS:

• Alpha divergence
• connection
• Fisher information metric
• maximum likelihood estimator
• Riemannian curvature.

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 60F99
• Secondary 62F12

Acknowledgment

We really appreciate the valuable comments of the reviewers, which improved the quality and readability of the paper.

Additional information

Funding

The author is supported by JSPS KAKENHI Grant Number 25380265.