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Statistical Science
2007, Vol. 22, No. 4, 598620DOI: 10.1214/07-STS249c Institute of
Mathematical Statistics, 2007
The Epic Story of Maximum LikelihoodStephen M. Stigler
Abstract. At a superficial level, the idea of maximum likelihood
mustbe prehistoric: early hunters and gatherers may not have used
the wordsmethod of maximum likelihood to describe their choice of
where andhow to hunt and gather, but it is hard to believe they
would have beensurprised if their method had been described in
those terms. It seemsa simple, even unassailable idea: Who would
rise to argue in favor ofa method of minimum likelihood, or even
mediocre likelihood? Andyet the mathematical history of the topic
shows this simple idea isreally anything but simple. Joseph Louis
Lagrange, Daniel Bernoulli,Leonard Euler, Pierre Simon Laplace and
Carl Friedrich Gauss areonly some of those who explored the topic,
not always in ways wewould sanction today. In this article, that
history is reviewed fromback well before Fisher to the time of
Lucien Le Cams dissertation.In the process Fishers unpublished 1930
characterization of conditionsfor the consistency and efficiency of
maximum likelihood estimates ispresented, and the mathematical
basis of his three proofs discussed.In particular, Fishers
derivation of the information inequality is seento be derived from
his work on the analysis of variance, and his laterapproach via
estimating functions was derived from Eulers Relationfor
homogeneous functions. The reaction to Fishers work is reviewed,and
some lessons drawn.
Key words and phrases: R. A. Fisher, Karl Pearson, Jerzy
Neyman,Harold Hotelling, Abraham Wald, maximum likelihood,
sufficiency, ef-ficiency, superefficiency, history of
statistics.
1. INTRODUCTION
In the 1860s a small group of young English intel-lectuals
formed what they called the X Club. Thename was taken as the
mathematical symbol for theunknown, and the plan was to meet for
dinner oncea month and let the conversation take them wherechance
would have it. The group included the Dar-winian biologist Thomas
Henry Huxley and the so-
Stephen M. Stigler is the Ernest DeWitt Burton
Distinguished Service Professor, Department of
Statistics, University of Chicago, Chicago, Illinois
60637, USA e-mail: [email protected].
This is an electronic reprint of the original articlepublished
by the Institute of Mathematical Statistics inStatistical Science,
2007, Vol. 22, No. 4, 598620. Thisreprint differs from the original
in pagination andtypographic detail.
cial philosopher-scientist Herbert Spencer. One eve-ning about
1870 they met for dinner at the AthenaeumClub in London, and that
evening included one ex-change that so struck those present that it
was re-peated on several occasions. Francis Galton was notpresent
at the dinner, but he heard separate ac-counts from three men who
were, and he recordedit in his own memoirs. As Galton reported it,
dur-ing a pause in the conversation Herbert Spencersaid, You would
little think it, but I once wrote atragedy. Huxley answered
promptly, I know thecatastrophe. Spencer declared it was
impossible,for he had never spoken about it before then.
Huxleyinsisted. Spencer asked what it was. Huxley replied,A
beautiful theory, killed by a nasty, ugly littlefact (Galton, 1908,
page 258).Huxleys description of a scientific tragedy is sin-
gularly appropriate for one telling of the history
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