Consistency in Concave Regression

Abstract

For each t in some subinterval T of the real line let Ft be a distribution function with mean m(t). Suppose m(t) is concave. Let t1, t2, ⋯ be a sequence of points in T and let Y1, Y2, ⋯ be an independent sequence of random variables such that the distribution function of Yk is Ftk . We consider estimators mn(t) = mn(t; Y1, ⋯, Yn) which are concave in t and which minimize n i=1 [ mn(ti; Y1, ⋯, Yn) - Yi]2 over the class of concave functions. We investigate their consistency and the convergence of mn'(t) to m'(t).