This research is devoted to the study of choice of the form of the statistical regression function in terms of the exogenous variables in the model when the number of exogenous variables is large. The common practice is to impose conditions such as linearity or additivity to avoid the difficulties associated with sparse data in high dimensions. Estimates of average derivative functionals will be used to select or reject a particular restriction on the form of the regression. The functionals considered are of integral type which may allow their estimation at the usual parametric rate. Estimators of these functionals which are based on kernel density estimators will be considered and their large sample properties analyzed. These estimators will then be used to test various hypotheses concerning the form of the regression function. The finite sample behavior of the proposed estimators and tests will be studied using constructed and real data. Under certain conditions these estimators will be studied for their use in identifying projection directions in projection pursuit regression. Similar functionals closely resembling the Fisher information matrix will be used to determine directions in projection pursuit density estimation.
