Goodness of fit test for isotonic regression

¹ Laboratoire de statistiques, bâtiment 425, Université Paris-Sud, 91405 Orsay Cedex, France; Cecile.Durot@math.u-psud.fr.
² Laboratoire Statistique et Génome, 523 place des Terrasses de l'Agora, 91000 Evry, et Département de Mathématiques, Université d'Evry-Val-d'Essonne, boulevard F. Mitterrand, 91025 Evry Cedex, France; atocquet@maths.univ-evry.fr.

Received: 15 December 1999
Revised: 5 September 2001

Abstract

We consider the problem of hypothesis testing within a monotone regression model. We propose a new test of the hypothesis H₀: “ƒ = ƒ₀” against the composite alternative H_a: “ƒ ≠ ƒ₀” under the assumption that the true regression function f is decreasing. The test statistic is based on the -distance between the isotonic estimator of f and the function f₀, since it is known that a properly centered and normalized version of this distance is asymptotically standard normally distributed under H₀. We study the asymptotic power of the test under alternatives that converge to the null hypothesis.