Asymptotical improvement of maximum likelihood estimators on Kullback–Leibler loss |
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Authors: | Shinto Eguchi Takemi Yanagimoto |
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Institution: | 1. Institute of Statistical Mathematics, Graduate University for Advanced Studies, Minami-Azabu, Minato-ku, Tokyo 106-8569, Japan;2. Graduate School of Science and Engineering, Chuo University, 1-13-27, Kasuga, Bunkyo-ku Tokyo 112-8551, Japan |
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Abstract: | We discuss the general form of a first-order correction to the maximum likelihood estimator which is expressed in terms of the gradient of a function, which could for example be the logarithm of a prior density function. In terms of Kullback–Leibler divergence, the correction gives an asymptotic improvement over maximum likelihood under rather general conditions. The theory is illustrated for Bayes estimators with conjugate priors. The optimal choice of hyper-parameter to improve the maximum likelihood estimator is discussed. The results based on Kullback–Leibler risk are extended to a wide class of risk functions. |
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Keywords: | 62F10 62F12 |
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