Estimating a Convex Function in Nonparametric Regression

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 L p -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.

Bounding quantile demand functions using revealed preference inequalities

“…Birke and Dette (2007), Mammen (1991), Mukerjee (1988): We …rst demonstrate that as n ! 1, the unrestricted estimator,d, satis…es RP almost surely.…”

“…Sinced C is a constrained version ofd, this implies thatd C =d w.p.a.1. Similar assumptions and proof strategies have been employed in Birke and Dette (2007) [ Mammen (1991)]: They assume that the function being estimated is strictly convex [monotone], such that the unconstrained estimator is convex [monotone] w.p.a.1. Since D T C ( ) D T C ( ), our new estimator will in general be less precise than the one de…ned as the optimizer over D T C ( ), but if the di¤erence > 0 is not too big, the additional estimation error should be negligible.…”

Optimal Admission and Scholarship Decisions: Choosing Customized Marketing Offers to Attract a Desirable Mix of Customers

“…Next we need to ensure the first equality in (12). By adding the following inequalities we properly model the average…”

“…11 In fact our inability to solve this problem to our satisfaction was the prime motivation for the development of the more complex procedure described in Section 2.5 and Appendices A.1 and A.2. 12 We estimated a simple binary logistic regression which included student characteristics and scholarship amount along with numerous interactions. In addition, we allowed for a flexible polynomial function of scholarships.…”

Nonparametric estimation and inference under shape restrictions

“…The literature on inference under monotonicity or convexity restrictions is vast. See, among many others, Birke and Dette (2006); Dumbgen (2003); Chernozhukov, Fernandez-Val, and Galichon (2009) ;Dette, Neumeyer, and Pilz (2006); Groeneboom, Jongbloed, and Wellner (2001); Pal and Woodroofe (2007); and the references therein. Existing results do not treat shape restrictions such as increasing or decreasing returns to scale and the Slutsky inequality that are of particular importance in economics.…”