Efficient prediction in L²-differentiable families of distributions

2013/12/12 by Emmanuel Onzon, Onzon, Emmanuel, université Lyon
Computer Science · Decision Sciences · Economics, Econometrics and Finance · Mathematics · #62J02 #62M20 #Bayesian Methods and Mixture Models #FOS: Mathematics #Probability and Risk Models #Risk and Portfolio Optimization #Statistical Distribution Estimation and Applications #Statistics Theory (math.ST) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1312.3625

openalex publication_date 2013/12/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A proof of the Cramér-Rao inequality for prediction is presented under conditions of L²-differentiability of the family of distributions of the model. The assumptions and the proof differ from those of Miyata (2001) who also proved this inequality under L²-differentiability conditions. It is also proved that if an efficient predictor (i.e. which risk attains the bound) exists then the family of distributions is of a special form which can be seen as an extension of the notion of exponential family. This result is also proved under L²-differentiability conditions.

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