Estimating a convex function in nonparametric regression
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Date
2005-07-29T09:23:21Z
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Abstract
A new nonparametric estimate of a convex regression function is proposed and its stochastic
properties are studied. The method starts with an unconstrained estimate of the derivative of
the regression function, which is firstly isotonized and then integrated. We prove asymptotic
normality of the new estimate and show that it is first order asymptotically equivalent to the
initial unconstrained estimate if the regression function is in fact convex. If convexity is not
present the method estimates a convex function whose derivative has the same Lp-norm as the
derivative of the (non-convex) underlying regression function. The finite sample properties of
the new estimate are investigated by means of a simulation study and the application of the
new method is demonstrated in two data examples.
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Keywords
Convexity, Nadaraya-Watson estimate, Nondecreasing rearrangement, Nonparametric regression, Order restricted inference