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 Electronic Journal of Statistics > Vol. 2 (2008) open journal systems 


Empirical likelihood based testing for regression

Ingrid Van Keilegom, Université catholique de Louvain; Institute of Statistics
César Sánchez Sellero, Universidad de Santiago de Compostela
Wenceslao González Manteiga, Universidad de Santiago de Compostela


Abstract
Consider a random vector $(X,Y)$ and let $m(x)=E(Y|X=x)$. We are interested in testing $H_0 : m in {cal M}_{Theta,{cal G}} = {gamma(cdot,theta,g) : theta in Theta, g in {cal G}}$ for some known function $gamma$, some compact set $Theta subset mathbb{IR}^p$ and some function set ${cal G}$ of real valued functions. Specific examples of this general hypothesis include testing for a parametric regression model, a generalized linear model, a partial linear model, a single index model, but also the selection of explanatory variables can be considered as a special case of this hypothesis. To test this null hypothesis, we make use of the so-called marked empirical process introduced by Diebolt (1995) and studied by Stute (1997) for the particular case of parametric regression, in combination with the modern technique of empirical likelihood theory in order to obtain a powerful testing procedure. The asymptotic validity of the proposed test is established, and its finite sample performance is compared with other existing tests by means of a simulation study.

AMS 2000 subject classifications: Primary 62E20; secondary 62F03, 62F05, 62F40, 62G08, 62G10.

Keywords: Marked empirical process, Model check for regression, Nonlinear regression, Partial linear model, Residuals.

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Keilegom, Ingrid Van, Sellero, César Sánchez, Manteiga, Wenceslao González, Empirical likelihood based testing for regression, Electronic Journal of Statistics, 2, (2008), 581-604 (electronic). DOI: 10.1214/07-EJS152.

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