Empirical likelihood based testing for regression

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 $IR$^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 \citeD and studied by \citeSt 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.Comment: Published in at http://dx.doi.org/10.1214/07-EJS152 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org

A review on empirical likelihood methods for regression

“…This result is important, as it shows that we can obtain empirical likelihood confidence regions for β 0 without estimating any variance. When the interest lies in testing the validity of the whole partial linear model by means of an EL approach (instead of testing only the value of the parameter vector β 0 ), one can use the method developed by Chen and Van Keilegom (2009) and Van Keilegom, Sánchez Sellero and González Manteiga (2008). In the former paper the authors developed a general smoothing based EL approach to test the validity of any semiparametric model, whereas the latter paper considers the same testing problem, but by using an EL approach based on marked empirical processes, which is quite different in nature from the former approach.…”

“…As for the partial linear model, the validity of the single-index model can be tested by using the tests developed by Chen and Van Keilegom (2009) and Van Keilegom, Sánchez Sellero and González Manteiga (2008). The above asymptotic results can also be obtained from Hjort, McKeague and Van Keilegom (2009), who developed generic conditions for the asymptotic theory of any EL ratio, built up using estimating equations depending on plug-in estimators of unknown nuisance parameters (see also Section 6.3).…”

“…Tripathi and Kitamura (2003) propose an EL test for conditional moment restrictions. For testing semiparametric regression models, Chen and Van Keilegom (2009) developed a smoothing based EL approach, whereas Van Keilegom, Sánchez Sellero and González Manteiga (2008) use an EL approach based on marked empirical processes.…”

Testing a parametric function against a non‐parametric alternative in IV and GMM settings

“…The GMM-ACH test uses the same setup as their GMM test, but where the latter is based on Hansen's J -statistic the GMM-ACH tests is based on a sequence of LM statistics. Other approaches to testing the validity of a conditional moment restriction consider tests based on a marked empirical process and Kolmogorov-Smirnov or Cramér-von Mises statistics, see Stute (1997), Andrews (1997), Whang (2001), and Van Keilegom et al (2008); and tests based on non-parametric estimation, see Tripathi and Kitamura (2003). 1 Finally, the test proposed by Horowitz (2006) has a form similar to the ICM test, but uses a particular class of density functions for weighting instead of exponential functions.…”

Testing a Parametric Function Against a Nonparametric Alternative in IV and GMM Settings

“…The GMM-ACH test uses the same setup as their GMM-test, but where the latter is based on Hansen's J-statistic the GMM-ACH tests is based on a sequence of LM statistics. Other approaches to testing the validity of a conditional moment restriction considers tests based on a marked empirical process and Kolmogorov-Smirnov or Cramér-von Mises statistics, see Stute (1997), Andrews (1997), Whang (2001), andVan Keilegom et al (2008), and tests based on nonparametric estimation, see Tripathi and Kitamura (2003). 1 Finally, the test proposed by Horowitz (2006) has a form similar to the ICM test, but uses a particular class of density functions for weighting instead of exponential functions.…”