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Biometrika 1993 80(4):715-730; doi:10.1093/biomet/80.4.715
© 1993 by Biometrika Trust
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A symbolic algorithm for studying adjustments to the profile likelihood

JAMES E. STAFFORD1 and DAVID F. ANDREWS2

1Department of Statistics, University of Oxford Oxford OX1 3TG, U.K.
2Department of Statistics, University of Toronto Toronto, Ontario, Canada M5S 1A1

A symbolic algorithm for the derivation of the asymptotic expansion of the maximum likelihood estimate was first introduced by Andrews & Stafford (1993). This algorithm is quite general and can be modified to give asymptotic expansions for the maximum likelihood estimate based on either the conditional, modified or adjusted profile likelihoods. Expansions are obtained quickly with no chance for human clerical error. As a result, these three adjustments can be easily studied and some new statistical results are obtained. Bartlett corrections of each adjusted likelihood, and asymptotic expansions for the first four moments of the maximum likelihood estimate based on the conditional and adjusted profile likelihoods, are derived.

Key Words: Adjusted likelihood • Asymptotic expansion • Bartlett correction • Profile likelihood • Symbolic computation


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