A New Proof of Fisher's Invariance Theorem Document ID: #026-RD-2015 Revision 1: 15/03/2019 19:42 Copyright 1997 – 2019: Zenith Genetica Ltd 1 A NEW PROOF OF FISHER’S INVARIANCE THEOREM Isaac Siwale Technical Report: #026-RD-2015 Zenith Genetica Limited London England e-mail: ike_siwale@hotmail.com Abstract This paper offers a new proof of Fisher’s invariance theorem. The proof is based on the definition of a likelihood function and it employs known results from probability and optimization, coupled with some elementary geometry of quadratic functions. Key Words: Maximum Likelihood, Invariance Theorem. 1. Introduction isher’s maximum likelihood method of parameter estimation includes a result on the invariance of the maximum likelihood estimator. Briefly stated, the ‘invariance theorem’ asserts that if ˆ is the maximum likelihood estimator of a parameter , then ) ˆ ( u is the maximum likelihood estimator of ) ( = u , where u is some function of . 1 In his 1922 paper, Fisher presents this invariance property by way of a specific example [4, pp.291-292]—he does not generalize beyond that particular instance. Following a fairly extensive review of the literature, Olive [5] observes that most books and papers merely state the invariance theorem without an accompanying proof, or they cite Zenha [9] whose main contribution is an extension of the applicability of the theorem beyond the proven case where u is a one-to-one function. Zenha achieves his extension by introducing an ‘induced likelihood’ function, viz.: = ) ( : ) | ( sup ) ( u L M d . (1) But others have since questioned Zenha’s method; in his review of Zenha’s paper, Berk [2] comments thus: “. . . when u is not one-to-one, M() in general appears not to be a likelihood function associated with any random variable. That maximizes M() is perhaps interesting but irrelevant to maximum likelihood estimation” [Paraphrased from 2] Berk offers his own (rather subtle and informal) proof of the invariance theorem by a simple recourse to simultaneous parameter estimation; granted the proven case when u is one-to-one, he asserts the following: “Justification for calling the maximum likelihood estimate of = (Φ) is implicit in the usual convention regarding simultaneous maximum likelihood estimation. If = (λ 1 ,λ 2 ) and = (λ 1 ,λ 2 ) is its maximum likelihood estimator, then it seems generally agreed that the maximum likelihood estimator of λ 1 = (λ) is λ 1 = (λ ) [. . .] the result [holds] for any u if one simply adjoins to u another function v so that the mapping → ((), ()) is one-to-one.” [Paraphrased from 2] The purpose of this paper is to propose a different and explicit proof of Fisher’s invariance theorem. The suggested proof is anchored on the definition of the likelihood function itself; on the geometry of quadratic functions; and on some well-known results in probability and optimization that are outlined in §2. The proof proper—including the assumptions upon which it is based—is presented in §3; §4 presents a summary of the paper, and §5 sets forth the legal framework governing its publication. 1 Other invariance properties of the maximum likelihood estimator—one of which follows directly from Fisher’s theorem—are in [8]. F