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TITLE: Constrained Maximum Likelihood Estimators for Densities: A Variational Perspective
ABSTRACT:
We present a framework for nonparametric density estimation in situations where the sample is supplemented by information and assumptions about shape, support, continuity, slope, location of modes, density values, etc. These supplements are incorporated as constraints that in conjunction
with a maximum likelihood criterion lead to constrained infinite-dimensional optimization problems that we formulate for the first time over spaces of semicontinuous functions. These spaces, when equipped with the hypo-distance, offer a series of advantages including simple conditions for existence of estimators and their limits and the guaranteed convergence of modes of densities. Relying on epi-convergence, we provide general conditions under which estimators subject to nearly arbitrary constraints are consistent and illustrate the framework with a number of examples that span classical and novel shape constraints.
BIO:Dr. Johannes O. Royset is Associate Chair of Research and Professor of Operations Research at the Naval Postgraduate School. Prof. Royset's research focuses on formulating and solving stochastic optimization and variational problems arising in data science, sensor management, and engineering design. He was awarded a National Research Council postdoctoral fellowship in 2003, a Young Investigator Award from the Air Force Office of Scientific Research in 2007, and the Barchi Prize as well as the MOR Journal Award from the Military Operations Research Society in 2009. He received the Carl E. and Jessie W. Menneken Faculty Award for Excellence in Scientific Research in 2010 and was a co-recipient of the UPS George D. Smith Prize from INFORMS in 2013. He was a plenary speaker at the 14th International Conference on Stochastic Programming (2016). Prof. Royset is a Guest Editor of Mathematical Programming and an Associate Editor of Operations Research, Naval Research Logistics, Journal of Optimization Theory and Applications, and Computational Optimization and Applications. In 2015-2016, he was a Guest Editor of Journal of Optimization Theory and Applications. His research has been supported by the Office of Naval Research, Air Force Office of Scientific Research, Army Research Office, and DARPA and has resulted in one book, three book chapters, and 45 journal publications. He has a Doctor of Philosophy degree from the University of California at Berkeley (2002).
