## Abstract

The Maximum Likelihood Estimator (MLE) is widely used in estimating information measures, and involves 'plugging-in' the empirical distribution of the data to estimate a given functional of the unknown distribution. In this work we propose a general framework and procedure to analyze the nonasymptotic performance of the MLE in estimating functionals of discrete distributions, under the worst-case mean squared error criterion. We show that existing theory is insufficient for analyzing the bias of the MLE, and propose to apply the theory of approximation using positive linear operators to study this bias. The variance is controlled using the well-known tools from the literature on concentration inequalities. Our techniques completely characterize the maximum L_{2} risk incurred by the MLE in estimating the Shannon entropy H(P) = σ_{i=1}^{S} -p_{i}ln p_{i}, and F_{α}(P) = σ_{i=1}^{S}p_{i}^{α} up to a multiplicative constant. As a corollary, for Shannon entropy estimation, we show that it is necessary and sufficient to have n ≪ S observations for the MLE to be consistent, where S represents the support size. In addition, we obtain that it is necessary and sufficient to consider n ≪ S^{1/α} samples for the MLE to consistently estimate F_{α}(P); 0 <α < 1. The minimax rate-optimal estimators for both problems require S/ln S and S^{1/α} / ln S samples, which implies that the MLE is strictly sub-optimal. When 1 < α < 3/2, we show that the maximum L_{2} rate of convergence for the MLE is n^{-2(α-1)} for infinite support size, while the minimax L_{2} rate is (n ln n)^{-2(α-1)}. When α ≥ 3/2, the MLE achieves the minimax optimal L_{2} convergence rate n^{-1} regardless of the support size.

Original language | English (US) |
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Title of host publication | Proceedings - 2015 IEEE International Symposium on Information Theory, ISIT 2015 |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 839-843 |

Number of pages | 5 |

ISBN (Electronic) | 9781467377041 |

DOIs | |

State | Published - Sep 28 2015 |

Event | IEEE International Symposium on Information Theory, ISIT 2015 - Hong Kong, Hong Kong Duration: Jun 14 2015 → Jun 19 2015 |

### Publication series

Name | IEEE International Symposium on Information Theory - Proceedings |
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Volume | 2015-June |

ISSN (Print) | 2157-8095 |

### Other

Other | IEEE International Symposium on Information Theory, ISIT 2015 |
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Country/Territory | Hong Kong |

City | Hong Kong |

Period | 6/14/15 → 6/19/15 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- Information Systems
- Modeling and Simulation
- Applied Mathematics