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Abstract:
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The celebrated, and much maligned, method of maximum likelihood is enjoying a modest revival for semiparametric models. For shape constrained density and regression problems, and for mixture models and compound decision problems nonparametric maximum likelihood offers an efficient and tuning-parameter-free computational strategy. Reweighted forms of quantile regression that borrow strength across nearby quantiles can also be formulated in likelihood terms. Several variants of these unusual likelihoods will be reviewed, stressing some open problems.
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