Testing in generalized partially linear models: A robust approach

Abstract

In this paper, we introduce a family of robust statistics which allow to decide between a parametric model and a semiparametric one. More precisely, under a generalized partially linear model, i.e., when the observations satisfy y i | (x i , t i) ∼ F (•, µ i) with µ i = H η(t i) + x t i β and H a known link function, we want to test H 0 : η(t) = α + γt against H 1 : η is a nonlinear smooth function. A general approach which includes robust estimators based on a robustified deviance or a robustified quasilikelihood is considered. The asymptotic behavior of the test statistic under the null hypothesis is obtained.

Robust testing to compare regression curves

“…For instance, Dette and Marchlewski (2010) considered a robust test for homoscedasticity in nonparametric regression. Under a partly linear regression model, Bianco et al (2006) proposed a test to study if the nonparametric component equals a fixed given function, while Boente et al (2013) considered the hypothesis that the nonparametric function is a linear function under a generalized partially linear model. On the other hand, Sun (2006), Dette et al (2011Dette et al ( , 2013 and Kuruwita et al (2014) considered the problem of comparing quantile functions.…”

Robust testing for superiority between two regression curves

“…Recently, Dette and Marchlewski (2010) considered a robust test for homoscedasticity in nonparametric regression. On the other hand, under a partly linear regression model, Bianco et al (2006) proposed a test to study if the nonparametric component equals a fixed given function, while Boente et al (2013) considered the hypothesis that the nonparametric function is a linear function under a generalized partially linear model. For the problem of testing superiority between two regression curves, Koul and Schick (1997) defined a family of covariate-matched statistics and derived its asymptotic behaviour under the null hypothesis and under root−n local alternatives.…”