Nonparametric model checks of single-index assumptions

摘要

Semiparametric single-index assumptions are convenient and widely useddimen\-sion reduction approaches that represent a compromise between theparametric and fully nonparametric models for regressions or conditional laws.In a mean regression setup, the SIM assumption means that the conditionalexpectation of the response given the vector of covariates is the same as theconditional expectation of the response given a scalar projection of thecovariate vector. In a conditional distribution modeling, under the SIMassumption the conditional law of a response given the covariate vectorcoincides with the conditional law given a linear combination of thecovariates. Several estimation techniques for single-index models are availableand commonly used in applications. However, the problem of testing thegoodness-of-fit seems less explored and the existing proposals still have somemajor drawbacks. In this paper, a novel kernel-based approach for testing SIMassumptions is introduced. The covariate vector needs not have a density andonly the index estimated under the SIM assumption is used in kernel smoothing.Hence the effect of high-dimensional covariates is mitigated while asymptoticnormality of the test statistic is obtained. Irrespective of the fixeddimension of the covariate vector, the new test detects local alternativesapproaching the null hypothesis slower than $n^{-1/2}h^{-1/4},$ where $h$ isthe bandwidth used to build the test statistic and $n$ is the sample size. Awild bootstrap procedure is proposed for finite sample corrections of theasymptotic critical values. The small sample performances of our test comparedto existing procedures are illustrated through simulations.