Location
Mathematics/Psychology : MP 422
Date & Time
September 5, 2014, 11:00 am – 12:00 pm
Description
Speaker
Dr. Johan Lim
Seoul National University
Title
Constrained Uniform Approximation and Estimation of Shape Restricted Functions.
Abstract
Thus far, a lot of work has been done on the two-stage estimator for shape-restricted functions that consists of nonparametric function estimation without taking into account shape constraints and shape modification via the constrained least squares approximation. This paper describes a new two-stage estimator for shape-restricted functions that can take into account the three shape constraints: monotonicity, convexity (concavity), and monotone convexity (concavity). In contrast with the previously known two-stage estimators, at the second stage, the proposed estimator uses the constrained uniform approximation instead of the constrained least squares approximation.
To support theoretically the proposed estimator, for each of the three shape constraints, we completely characterize the best uniform approximation of a function to the class of the corresponding shape-restricted functions. In particular, for the estimation of convex and monotone convex functions, the use of uniform approximation allows us to apply the computationally efficient convex hull algorithm--which can find the convex hull of n points in a plane in O(n log n) operations--to the projection problem at the second stage. This results in a much faster estimation algorithm than the previously known ones based on the constrained least squares approximation.
Since the class of shape-restricted functions considered here is much broader than the one dealt with in the prior work, direct comparison in the convergence rate is not made here, but the convergence rate analysis performed in this paper sheds light on its theoretical efficiency. Indeed, when the regressogram estimate is employed as the nonparametric estimate at the first stage, the proposed two-stage estimator is shown to converge to the true shape-restricted function at least constant times faster than the nonparametric estimate at the first stage. This is joint work with Seung-Jean Kim at Citi Capital Advisors at NY.
Abstract
Thus far, a lot of work has been done on the two-stage estimator for shape-restricted functions that consists of nonparametric function estimation without taking into account shape constraints and shape modification via the constrained least squares approximation. This paper describes a new two-stage estimator for shape-restricted functions that can take into account the three shape constraints: monotonicity, convexity (concavity), and monotone convexity (concavity). In contrast with the previously known two-stage estimators, at the second stage, the proposed estimator uses the constrained uniform approximation instead of the constrained least squares approximation.
To support theoretically the proposed estimator, for each of the three shape constraints, we completely characterize the best uniform approximation of a function to the class of the corresponding shape-restricted functions. In particular, for the estimation of convex and monotone convex functions, the use of uniform approximation allows us to apply the computationally efficient convex hull algorithm--which can find the convex hull of n points in a plane in O(n log n) operations--to the projection problem at the second stage. This results in a much faster estimation algorithm than the previously known ones based on the constrained least squares approximation.
Since the class of shape-restricted functions considered here is much broader than the one dealt with in the prior work, direct comparison in the convergence rate is not made here, but the convergence rate analysis performed in this paper sheds light on its theoretical efficiency. Indeed, when the regressogram estimate is employed as the nonparametric estimate at the first stage, the proposed two-stage estimator is shown to converge to the true shape-restricted function at least constant times faster than the nonparametric estimate at the first stage. This is joint work with Seung-Jean Kim at Citi Capital Advisors at NY.
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