We study nonparametric maximum likelihood estimation of a log-concave density function f0 which is known to satisfy further constraints, where either (a) the mode m of f0 is known, or (b) f0 is known to be symmetric about a fixed point m. We develop asymptotic theory for both constrained log-concave maximum likelihood estimators (MLE’s), including consistency, global rates of convergence, and local limit distribution theory. In both cases, we find the MLE’s pointwise limit distribution at m (either the known mode or the known center of symmetry) and at a point x0≠m. Software to compute the constrained estimators is available in the R package logcondens.mode. The symmetry-constrained MLE is particularly useful in contexts of location estimation. The mode-constrained MLE is useful for mode-regression. The mode-constrained MLE can also be used to form a likelihood ratio test for the location of the mode of f0. These problems are studied in separate papers. In particular, in a separate paper we show that, under a curvature assumption, the likelihood ratio statistic for the location of the mode can be used for hypothesis tests or confidence intervals that do not depend on either tuning parameters or nuisance parameters.
Bibliographical noteFunding Information:
∗Supported in part by NSF Grants DMS-1104832 and a University of Minnesota Grant-In-Aid grant. †Supported in part by NSF Grants DMS-1104832 and DMS-1566514, and NI-AID grant 2R01 AI291968-04.
Both authors owe thanks to Tilmann Gneiting for support during visits to the Applied Mathematics Institute at the University of Heidelberg in 2011-2012. We also owe thanks to Lutz D?mbgen for several helpful conversations. We are grateful to an Associate Editor and two referees for a very careful reading and for helpful comments.
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- Convergence rate
- Convex optimization
- Empirical processes
- Shape constraints