Computational Algebraic Methods in Efficient Estimation
A strong link between information geometry and algebraic statistics is made by investigating statistical manifolds which are algebraic varieties. In particular it is shown how first and second order efficient estimators can be constructed, such as bias corrected Maximum Likelihood and more general estimators, and for which the estimating equations are purely algebraic. In addition it is shown how Gröbner basis technology, which is at the heart of algebraic statistics, can be used to reduce the degrees of the terms in the estimating equations. This points the way to the feasible use, to find the estimators, of special methods for solving polynomial equations, such as homotopy continuation methods. Simple examples are given showing both equations and computations.
KeywordsEfficient Estimator Exponential Family Algebraic Statistic Information Geometry Asymptotic Efficiency
This paper has benefited from conversations with and advice from a number of colleagues. We should thank Satoshi Kuriki, Tomonari Sei, Wicher Bergsma and Wilfred Kendall. The first author acknowledges support by JSPS KAKENHI Grant 20700258, 24700288 and the second author acknowledges support from the Institute of Statistical Mathematics for two visits in 2012 and 2013 and from UK EPSRC Grant EP/H007377/1. A first version of this paper was delivered at the WOGAS3 meeting at the University of Warwick in 2011. We thank the sponsors. The authors also thank the referees of the short version in GSI2013 and the referees of the first long version of the paper for insightful suggestions.
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