Statistical Science

On the history of maximum likelihood in relation to inverse probability and least squares

Anders Hald

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Abstract

It is shown that the method of maximum likelihood occurs in rudimentary forms before Fisher [Messenger of Mathematics 41 (1912) 155–160], but not under this name. Some of the estimates called “most probable” would today have been called “most likely.” Gauss [Z. Astronom. Verwandte Wiss. 1 (1816) 185–196] used invariance under parameter transformation when deriving his estimate of the standard deviation in the normal case. Hagen [Grundzüge der Wahrschein­lichkeits­Rechnung, Dümmler, Berlin (1837)] used the maximum likelihood argument for deriving the frequentist version of the method of least squares for the linear normal model. Edgeworth [J. Roy. Statist. Soc. 72 (1909) 81–90] proved the asymptotic normality and optimality of the maximum likelihood estimate for a restricted class of distributions. Fisher had two aversions: noninvariance and unbiasedness. Replacing the posterior mode by the maximum likelihood estimate he achieved invariance, and using a two­stage method of maximum likelihood he avoided appealing to unbiasedness for the linear normal model.

Article information

Source
Statist. Sci. Volume 14, Number 2 (1999), 214-222.

Dates
First available in Project Euclid: 24 December 2001

Permanent link to this document
http://projecteuclid.org/euclid.ss/1009212248

Digital Object Identifier
doi:10.1214/ss/1009212248

Mathematical Reviews number (MathSciNet)
MR1722061

Zentralblatt MATH identifier
02068904

Citation

Hald, Anders. On the history of maximum likelihood in relation to inverse probability and least squares. Statist. Sci. 14 (1999), no. 2, 214--222. doi:10.1214/ss/1009212248. http://projecteuclid.org/euclid.ss/1009212248.


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