Open Access
VOL. 10 | 2013 Asymptotics of Maximum Likelihood without the LLN or CLT or Sample Size Going to Infinity
Charles J. Geyer

Editor(s) Galin Jones, Xiaotong Shen

Inst. Math. Stat. (IMS) Collect., 2013: 1-24 (2013) DOI: 10.1214/12-IMSCOLL1001


If the log likelihood is approximately quadratic with constant Hessian, then the maximum likelihood estimator (MLE) is approximately normally distributed. No other assumptions are required. We do not need independent and identically distributed data. We do not need the law of large numbers (LLN) or the central limit theorem (CLT). We do not need sample size going to infinity or anything going to infinity.

Presented here is a combination of Le Cam style theory involving local asymptotic normality (LAN) and local asymptotic mixed normality (LAMN) and Cramér style theory involving derivatives and Fisher information. The main tool is convergence in law of the log likelihood function and its derivatives considered as random elements of a Polish space of continuous functions with the metric of uniform convergence on compact sets. We obtain results for both one-step-Newton estimators and Newton-iterated-to-convergence estimators.


Published: 1 January 2013
First available in Project Euclid: 23 September 2013

zbMATH: 1327.62123
MathSciNet: MR3586936

Digital Object Identifier: 10.1214/12-IMSCOLL1001

Primary: 60F05 , 62F12
Secondary: 62F40

Keywords: locally asymptotically normal , maximum likelihood , Newton’s method , no-$n$ asymptotics , Parametric bootstrap , quadraticity

Rights: Copyright © 2013, Institute of Mathematical Statistics

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