Empirical likelihood based testing for regression

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

The bootstrap -A review

The bootstrap, extensively studied during the last decade, has become a powerful tool in different areas of Statistical Inference. In this work, we present the main ideas of bootstrap methodology in several contexts, citing the most relevant contributions and illustrating with examples and simulation studies some interesting aspects

Significance testing in nonparametric regression base on the bootstrap

We propose a test for selecting explanatory variables in nonparametric regression. The test does not need to estimate the conditional expectation function given all the variables but only those which are significant under the null hypothesis. This feature is compntationally convenient and solves, in part, the problem of the "curse of dimensionality" when selecting regressors in a nonparametric context. The proposed test statistic is based on functionals of an empirical process marked by nonparametric residuals. Contiguous alternatives, converging to the null at a rate n-1I2 can be detected. The asymptotic null distribution of the statistic depends on certain features of the data generating process, and asymptotic tests are difficult to implement except in rare circumstances. We justify the consistency of two bootstrap tests easy to implement, which exhibit good level accuracy for fairly small samples, according to the Monte Carlo simulations reported. These results are also applicable to test other interesting restrictions on nonparametric regression curves, like partial linearity and conditional independence

Comments on: Nonparametric estimation in mixture cure models with covariates

This work was supported by the project PID2020-116587GB-I00 funded by MCIN/AEI/10.13039/501100011033 and the Competitive Reference Groups 2021/2024 (ED431C2021/24) from the Xunta de Galicia. Open Access funding provided thanks to the CRUE-CSIC agreement with Springer NatureS