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Could anyone give me some information on who invented profile maximum likelihood estimation or who first use profile maximum likelihood estimation and the short history of profile maximum likelihood estimation? I would like to know about any paper on this.

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    $\begingroup$ I would say that this is in Fisher itself, see here. $\endgroup$ – tchakravarty Nov 19 '12 at 7:49
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    $\begingroup$ Excellent catch @fg nu - I had thought it was Sprott and Kalbfleisch but obviously Sprott would know better. All lot of potential confusion can be avoided by attention to distinguishing just the maximum from the whole function - it is the whole function where profiling leads to the biggest change. If you moved this up to an answer, I'd vote for it. $\endgroup$ – phaneron Nov 19 '12 at 16:49
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I believe that the first use of the concept of profile likelihoods is in Fisher's own work, as explained here.

The relevant quote from that link to the book by D. A. Sprott is:

Profile likelihoods have existed for a long time in the form of likelihood ratio tests. But these are not used as likelihood functions to produce likelihood inferences. The first use of a profile likelihood to produce likelihood inferences appears to be Fisher's 1956 "relative likelihood of the two-by-two table", Fisher (1991c, p. 136). His likelihood (101) is the profile likelihood of $p_1 = p_2$ for the $2\times2$ table formed by the two binomial distributions (Example 2.9.13). Box and Cox (1964) graphed and used the profile likelihood under the name maximized likelihood, but mainly for obtaining the maximum likelihood estimate and to establish the confidence intervals using the $\chi^2$ likelihood ratio procedure. Sprott and Kalbfleisch (1965) used the profile likelihood without giving it a name, to produce likelihood inferences. Sprott and Kalbfleisch (1969) gave further examples and graphs under the name maximum relative likelihood.

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