A modified closed-form maximum likelihood estimator

The maximum likelihood estimator plays a fundamental role in statistics. However, for many models, the estimators do not have closed-form expressions. This limitation can be significant in situations where estimates and predictions need to be computed in real-time, such as in applications based on embedded technology, in which numerical methods can not be implemented. This paper provides a modification in the maximum likelihood estimator that allows us to obtain the estimators in closed-form expressions under some conditions. Under mild conditions, the estimator is invariant under one-to-one transformations, consistent, and has an asymptotic normal distribution. The proposed modified version of the maximum likelihood estimator is illustrated on the Gamma, Nakagami, and Beta distributions and compared with the standard maximum likelihood estimator.

[1]  R. Redner.  Note on the Consistency of the Maximum Likelihood Estimate for Nonidentifiable Distributions , 1981 .

[2]  C. Gouriéroux,et al.  PSEUDO MAXIMUM LIKELIHOOD METHODS: THEORY , 1984 .

[3]  F. Louzada,et al.  A Note on Bias of Closed-Form Estimators for the Gamma Distribution Derived From Likelihood Equations , 2019, The American Statistician.

[4]  E. B. Andersen,et al.  Asymptotic Properties of Conditional Maximum‐Likelihood Estimators , 1970 .

[5]  J. Hosking.  L‐Moments: Analysis and Estimation of Distributions Using Linear Combinations of Order Statistics , 1990 .

[6]  .. W. V. Der,et al.  On Profile Likelihood , 2000 .

[7]  Stephen E. Fienberg,et al.  Testing Statistical Hypotheses , 2005 .

[8]  D. Firth.  Bias reduction of maximum likelihood estimates , 1993 .

[9]  J. Anderson,et al.  Penalized maximum likelihood estimation in logistic regression and discrimination , 1982 .

[10]  Francisco Louzada,et al.  Bias Reduction in the Closed-Form Maximum Likelihood Estimator for the Nakagami-m Fading Parameter , 2020, IEEE Wireless Communications Letters.

[11]  Russell C. H. Cheng,et al.  Estimating Parameters in Continuous Univariate Distributions with a Shifted Origin , 1983 .

[12]  John H. K. Kao.  A Graphical Estimation of Mixed Weibull Parameters in Life-Testing of Electron Tubes , 1959 .

[13]  E. L. Lehmann,et al.  Theory of point estimation , 1950 .

[14]  J. Aldrich.  R.A. Fisher and the making of maximum likelihood 1912-1922 , 1997 .

[15]  John H. K. Kao.  Computer Methods for Estimating Weibull Parameters in Reliability Studies , 1958 .

[16]  Lee J. Bain,et al.  Inferential Procedures for the Generalized Gamma Distribution , 1970 .

[17]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .