  We develop inference and testing procedures for conditional dispersion and
skewness in a nonparametric regression setup based on statistical depth
functions. The methods developed can be applied in situations, where the
response is multivariate and the covariate is a random element in a metric
space. This includes regression with functional covariate as a special case. We
construct measures of the center, the spread and the skewness of the
conditional distribution of the response given the covariate using depth based
nonparametric regression procedures. We establish the asymptotic consistency of
those measures and develop a test for heteroscedasticity and a test for
conditional skewness. We present level and power study for the tests in several
simulated models. The usefulness of the methodology is also demonstrated in a
real dataset. In that dataset, our responses are the nutritional contents of
different meat samples measured by their protein, fat and moisture contents,
and the functional covariate is the absorbance spectra of the meat samples.
