Research Article

Goodness of fit test for isotonic regression

Durot, Cécile^a1 and Tocquet, Anne-Sophie^a2

^a1 Laboratoire de statistiques, bâtiment 425, Université Paris-Sud, 91405 Orsay Cedex, France; Cecile.Durot@math.u-psud.fr.

^a2 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.

Abstract

We consider the problem of hypothesis testing within a monotone regression model. We propose a new test of the hypothesis H ₀: “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 f ₀, since it is known that a properly centered and normalized version of this distance is asymptotically standard normally distributed under H ₀. We study the asymptotic power of the test under alternatives that converge to the null hypothesis.

(Received December 15 1999)

(Revised September 5 2001)

(Online publication August 15 2002)

Key Words:

• Nonparametric regression;
• isotonic estimator;
• goodness of fit test;
• asymptotic power.

Mathematics Subject Classification:

• 62G08;
• 62G10;
• 62G20