Constrained maximum likelihood estimators for densities

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dc.contributor.authorRoyset, Johannes O.
dc.contributor.authorWets, Roger J.-B.
dc.contributor.corporateNaval Postgraduate School (U.S.)en_US
dc.contributor.departmentOperations Research (OR)en_US
dc.date.accessioned2018-01-16T19:04:43Z
dc.date.available2018-01-16T19:04:43Z
dc.date.issued2017-06-09
dc.description.abstractWe propose a framework for nonparametric maximum likelihood estimation of densities in situations where the sample is supplemented by information and assumptions about shape, support, continuity, slope, location of modes, density values, and other conditions that, individually or in combination, restrict the family of densities under consideration. We establish existence of estimators and their cluster points, strong consistency under mild assumptions, and robustness in the presence of model misspecification. The results are achieved by means of viewing densities as elements of spaces of semicontinuous functions with the hypo-distance metric. This metric emerges as natural and convenient when considering broad classes of side conditions. It also has the exceptional property that convergence of densities in this metric implies convergence of modes, near-modes, height of modes, and high-likelihood events. Thus, we automatically achieve strong consistency of a rich class of plug-in estimators for modes and related quantities. Relying on almost sure epi-convergence of criterion functions, we avoid the strong assumptions associated with uniform laws of large numbers and instead leverage a less demanding law, for which we provide a new proof. Specific examples illustrate the framework including an estimator simultaneously subject to bounds on density values and its (sub)gradients, restriction to concavity, penalization that encourages lower modes, and imprecise information about the expected value.en_US
dc.format.extent35 p.en_US
dc.identifier.citationJ.O. Royset, R.J.-B. Wets, "Constrained maximum likelihood estimators for densities," arXic:1702.08109v3 [math.ST] 9 Jun 2017en_US
dc.identifier.urihttps://hdl.handle.net/10945/56666
dc.rightsThis publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. Copyright protection is not available for this work in the United States.en_US
dc.subject.authorNonparametric density estimationen_US
dc.subject.authorMaximum likelihooden_US
dc.subject.authorShape-constrained estimationen_US
dc.subject.authorConsistencyen_US
dc.subject.authorVariational approximationsen_US
dc.subject.authorEpi-convergenceen_US
dc.subject.authorHypo-distanceen_US
dc.subject.authorEpi-splinesen_US
dc.titleConstrained maximum likelihood estimators for densitiesen_US
dc.typeArticleen_US
dspace.entity.typePublication
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