  Semiparametric single-index assumptions are convenient and widely used
dimen\-sion reduction approaches that represent a compromise between the
parametric and fully nonparametric models for regressions or conditional laws.
In a mean regression setup, the SIM assumption means that the conditional
expectation of the response given the vector of covariates is the same as the
conditional expectation of the response given a scalar projection of the
covariate vector. In a conditional distribution modeling, under the SIM
assumption the conditional law of a response given the covariate vector
coincides with the conditional law given a linear combination of the
covariates. Several estimation techniques for single-index models are available
and commonly used in applications. However, the problem of testing the
goodness-of-fit seems less explored and the existing proposals still have some
major drawbacks. In this paper, a novel kernel-based approach for testing SIM
assumptions is introduced. The covariate vector needs not have a density and
only the index estimated under the SIM assumption is used in kernel smoothing.
Hence the effect of high-dimensional covariates is mitigated while asymptotic
normality of the test statistic is obtained. Irrespective of the fixed
dimension of the covariate vector, the new test detects local alternatives
approaching the null hypothesis slower than $n^{-1/2}h^{-1/4},$ where $h$ is
the bandwidth used to build the test statistic and $n$ is the sample size. A
wild bootstrap procedure is proposed for finite sample corrections of the
asymptotic critical values. The small sample performances of our test compared
to existing procedures are illustrated through simulations.
