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RoseSmithD

# RoseSmithD - The Statistician(2000 49 Part 2 pp 229240...

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Symbolic maximum likelihood estimation with Mathematica Colin Rose Theoretical Research Institute, Sydney, Australia and Murray D. Smith University of Sydney, Australia [Received May 1998. Revised October 1999] Summary. Mathematica is a symbolic programming language that empowers the user to undertake complicated algebraic tasks. One such task is the derivation of maximum likelihood estimators, demonstrably an important topic in statistics at both the research and the expository level. In this paper, a Mathematica package is provided that contains a function entitled SuperLog . This function utilizes pattern-matching code that enhances Mathematica’s ability to simplify expressions involving the natural logarithm of a product of algebraic terms. This enhancement to Mathematica’s func- tionality can be of particular benefit for maximum likelihood estimation. Keywords : Computer algebra systems; Estimate; Estimator; Mathematica; Symbolic maximum likelihood; Teaching 1. Introduction Although statistical software has long been used for maximum likelihood (ML) estimation, the focus of attention has almost always been on obtaining ML estimates (a numerical problem), ra- ther than on deriving ML estimators (a symbolic problem). Even after the introduction of powerful computer algebra systems, in the context of ML estimation, symbolic engines have largely only been used to solve numerical problems (see, for example, Currie (1995) and Cook and Broemeling (1995)) or to derive large sample symbolic approximations to ML estimators (see Stafford and Andrews (1993) and Stafford et al. (1994)). By contrast, we use a computer algebra system for the first time to derive exact symbolic ML estimators from first principles. This paper shows how Mathematica (see Wolfram (1996)) can be extended to handle symbolic ML estimation, which is demonstrably an important topic in statistics, at both the research and the expository level. The paper has five sections. Section 2 expands Mathematica’s programming language to handle symbolic ML estimation. Section 3 illustrates the approach with three simple expository examples and is therefore well suited for teaching purposes. Section 4 extends the analysis to more challenging material. Section 5 concludes. Appendices follow giving a glossary of Mathematica terms and the package code. & 2000 Royal Statistical Society 0039–0526/00/49229 The Statistician (2000) 49 , Part 2, pp. 229–240 Address for correspondence : Murray D. Smith, Econometrics and Business Statistics, University of Sydney, Sydney, NSW 2006, Australia. E-mail: [email protected]

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2. Extending Mathematica: the package SMLE Consider the following simple problem. Let ( Y 1 , . . . , Y n ) denote a random sample of size n collected on Y ± Poisson( L ), where parameter L . 0 is unknown—show that the mean of the random sample is the ML estimator of L .
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