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Maximum likelihood methods may be inadequate for parameter estimation in models where many nuisance parameters are present. The modified profile likelihood (MPL) of Barndorff-Nielsen (1983) serves as a highly accurate approximation to the marginal or conditional likelihood, when either exist, and can be viewed as an approximate conditional likelihood when they do not. We examine the modified profile likelihood, its variants, and its connections with Laplace and saddlepoint approximations under both theoretical and pragmatic lenses.
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