Title: On Nonparametric Regression using Data Depth,
Abstract: We investigate nonparametric regression methods based on statistical depth
functions. These nonparametric regression procedures can be used 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. Our objective is to study different features of the conditional
distribution of the response given the covariate. We construct measures of the
center and the spread of the conditional distribution using depth based
nonparametric regression procedures. We establish the asymptotic consistency of
those measures and develop a test for heteroscedasticity based on the measure
of conditional spread. The usefulness of the methodology is demonstrated in
some real datasets. In one dataset consisting of Italian household expenditure
data for the period 1973 to 1992, we regress the expenditure for different
items on their prices. In another 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.