On the $\mathbb{L}_{p}$-error of monotonicity constrained estimators

Abstract

We aim at estimating a function λ : [0, 1] → R, subject to the constraint that it is decreasing (or increasing). We provide a unified approach for studying the Lp-loss of an estimator defined as the slope of a concave (or convex) approximation of an estimator of a primitive of λ, based on n observations. Our main task is to prove that the Lp-loss is asymptotically Gaussian with explicit (though unknown) asymptotic mean and variance. We also prove that the local Lp-risk at a fixed point and the global Lp-risk are of order n −p/3 . Applying the results to the density and regression models, we recover and generalize known results about Grenander and Brunk estimators. Also, we obtain new results for the Huang-Wellner estimator of a monotone failure rate in the random censorship model, and for an estimator of the monotone intensity function of an inhomogeneous Poisson process.

Goodness-of-Fit Test for Monotone Functions

“…against the non-parametric alternative that is decreasing, where ⊂ R r is a given set and for every , is a given decreasing function on [0, 1]. The criterion we consider for testing relies on the Grenander-type estimator introduced in Durot (2007). Precisely, we assume in the sequel that we have at hand a cadlag step estimator…”

“…whereˆ n is a suitable estimator of under H 0 and p is a fixed real. It should be mentioned that in the particular case of a simple null hypothesis (of the form H 0 : = 0 ), under H 0 , S pn is the L p -error ofˆ n so its asymptotic distribution is given by theorem 2 of Durot (2007); our main task here is to generalize the method of Durot (2007) to the case of a possibly composite null hypothesis. We need some notation to describe the asymptotic distribution of S pn under H 0 .…”

“…Similar to Durot's theorem 2, the assumptions that s > 3/4 and q > 12 could probably be weakened at the price of more technicalities. A sufficient condition for assumption (C) to hold is given by lemma 1 in Durot (2007). Assumptions (A), (B) and (C) allow computing the asymptotic distribution of the L p -error ofˆ n under H 0 , whereas assumptions (D) and (E) are designed to ensure that ˆ n properly estimates under H 0 .…”

“…Then, theorem 6 of Durot (2007) shows that (B) and (C) hold with L = (as there is no nuisance parameter here, we omit the subscript in the notation). SettingL n (1) = 1 and l n = ˆ n , it follows from (8) and corollary 1 that the test with critical region (5) has asymptotic level .…”

“…Huang & Wellner (1995) study a Grenander-type estimator for a monotone hazard rate in a right-censoring model. Reboul (2005) and Durot (2007) study such estimators in a general model that covers the cases of density, regression and right-censoring models: Reboul (2005) provides a non-asymptotic control for the L 1 -risk and proves that the estimator is spatially adaptive, whereas Durot (2007) gives the limit distribution of the L p -error as the number of observations tends to infinity.…”