An Omnibus Test of Goodness-of-fit for Conditional Distribution Functions with Applications to Regression Models

Gilles R. Ducharme (1) , Sandie Ferrigno (2)

1 EPS - Probabilités et statistiques
2 BIGS - Biology, genetics and statistics

We introduce a goodness-of-fit test for statistical models about the conditional distribution function of a random variable. This test is useful for assessing whether a regression model fits a data set regarding all assumptions made in its elaboration. The test is based on a generalization of the Cramer-von Mises statistic and involves a local polynomial estimator of the conditional distribution function. First, the uniform almost sure consistency of this estimator is established. Then, the asymptotic distribution of the test statistic is derived under the null hypothesis and local alternatives. The extension to the case where unknown parameters appear in the model is developed. Finally, a simulation study is performed to see how the test behaves with moderate samples. It emerges that, although the test can detect any departure from the null model, its power is comparable to that of other nonparametric tests designed to examine only specific departures.

Mots clés: Cramer-von Mises Statistic Goodness-of-fit test; Conditional distribution function; Local polynomial estimation; Regression Model

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Gilles R. Ducharme, Sandie Ferrigno. An Omnibus Test of Goodness-of-fit for Conditional Distribution Functions with Applications to Regression Models. Journal of Statistical Planning and Inference, 2012, 142 (10), pp.2748-2761. ⟨10.1016/j.jspi.2012.04.008⟩. ⟨hal-00641034⟩