Diego Colombo: Goodness of fit test for isotonic regression

Adviser: Marloes Maathuis

July 2008

Abstract

We study the work of Durot and Tocquet (2001), whom proposed a new test of the hypothesis H0 : ”f = f0” versus the composite alternative Hn : ”f != f0”, under the assumption that the true regression function f is monotone decreasing on [0, 1]. The test statistic is based on the L1-distance between the isotonic estimator ˆ fn of f and the given function f0, since a centered and normalized version of this distance, is asymptotically standard normal distributed under the null hypothesis H0, provided that the given function f0 satisfies some regularity conditions. The main purpose to study asymptotic normality of the isotonic estimator, relies on the study of its asymptotic power under the alternative Hn : ”f = f0 + cn"n”. The idea is to study the minimal rate of convergence for cn, such that the test has a prescribed asymptotic power. Durot and Tocquet show that this minimal rate is n−5/12 if "n does not depend on n and n−3/8 if it does.
Our contribution is a more detailed explanation of the models, of the main results and the insertion of some extra particular steps in the proofs. To check these theoretical results in simulations like Durot and Tocquet, we write new R codes. Namely, we perform a simulation study to compare the power of this test with that of the likelihood ratio test, for the case where f0 is linear, and we also compare these simulations results to the ones obtained by Durot and Tocquet. Moreover, we propose extra simulations for the power of another test not treated by Durot and Tocquet and we will see that it is always most powerful than the one they studied. Finally, we conduct a new simulation study in the case where the given monotone function f0 is quadratic.

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