Inference in projection pursuit regression.

Abstract

Projection pursuit is a data analytic tool that explores interesting nonlinear structures in multi-dimensional data through low dimensional projections of the data. It is an attractive technique especially when we have little information about which model is a good one for the data. This thesis concerns inference problems in PP regression: namely, estimation of the regression function and related hypothesis testing problems. In the hypothesis testing problem, we test the null hypothesis that the regression function is constant. The p-value associated with our test statistic indicates how extreme the observed value is. In theory, the p-value of our test statistic is sum of a few terms that have certain geometrical meanings. Johansen and Johnstone (1990) carried out a one-term approximation to the p-value of the test statistic in the rather restrictive setting where the predictors are standard multivariate normal. We have improved their procedure using a two term approximation based on a theorem (Sun, 1993) that justifies the relevance of the two term approximation under certain regularity conditions. We show that our procedure is applicable when the predictors have an arbitrary multivariate normal distribution. We extend our procedure to more generalized circumstances which broaden its applicability to real life data analysis situations. The family of Elliptically Contoured densities includes a comprehensive set of symmetric densities. We carry out the p-value computation when the predictors belong to the family of Elliptically Contoured densities. In the estimation problem, a projection pursuit regression fitting procedure is developed for the general predictor case. For this purpose, we construct an appropriate orthonormal basis with respect to the density of the predictor where the density is arbitrary. We developed approximation formula for the significance test when the predictors are EC-variates. As far as estimation is concerned, the density of the predictor is not constrained to be in the family Elliptically Contoured densities. Practical aspects concerning implementation are important issues. We investigate the effect of robust estimators of the location vector and dispersion matrix on the performance of PPR fits using the MVE (Minimum Volume Ellipsoid method) estimators. We also compare the performance of a new sphering method and a classical one using small samples when the covariance matrix of the predictor is ill-conditioned.