STATISTICAL INFERENCE UNDER ORDER RESTRICTIONS LIMIT THEORY FOR THE GRENANDER ESTIMATOR UNDER ALTERNATIVE HYPOTHESES
Jon A. Wellner · 2007
1. Limit theory for the Grenander estimator. Our goal in this section is to reprove some limit theory for the Grenander estimator. In particular we want to study limit distributions for rn( b fn(x0) f(x0) under the following hypotheses: Case 1. f(x) = 1[0;1](x); uniform density (or degenerate mixing distribution) Case 2. At a point x0 with f 0 (x0) < 0 Case 3. At a point x0 with f (j) (x0) = 0, j = 1;:::;k 1, f (k) (x0)6 0. Case 4. At a point x02 (a;b) with f(x) constant on (a;b). Case 5. At a point x0 where f is discontinuous. 1.1. Case 1: the Grenander estimator under sampling from a uniform density. Suppose that X1;:::;Xn are i.i.d. with a monotone densityf on (0;1). Then the Grenander estimator ^ fn off is the left-derivative of the least concave majorant of the empirical distribution function Fn. When f is the uniform density f(x) = 1[0;1](x), corresponding to a degenerate mixing distribution G in the representation of f as a scale mixture of uniform densities, Groeneboom and Pyke (1983) and Groeneboom (1985) showed that for each 0 < x0 < 1 p n( b fn(x0) f(x0))!d S(x0) (1) where S is the left derivative of the least concave majorant C of a standard Brownian bridge process U on [0; 1]. The properties of the limiting processes C and S have been studied by Groeneboom (1983), Pitman (1983), Bass (1984), C