{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/87422"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/87422","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Building a Nonparametric Model After Dimension Reduction","abstract":"To effectively build a regression model with a large number of covariates is no easy task. We consider using dimension reduction before building a parametric or spline model. The dimension reduction procedure is based on a canonical correlation analysis on the predictor variables and a spline basis generated for the response variable. One important question in dimension reduction is to decide on the number of effective dimensions needed. We study four tests of dimensionality: a chi-square test, a Wald-type test on eigenvalues, a modified Wald-type test, and a matrix rank test. These tests are motivated from different aspects of the problem and have their own strength and weakness. We discuss and compare these tests both theoretically and through Monte Carlo simulations, based on which specific recommendations for determining dimensionality are made. Additive regression splines are first fitted to the data in the space of reduced dimensionality. A Tukey-type test of additivity is proposed and compared with Rao's score test. When the hypothesis of additivity is rejected, tensor product splines can be used for model building.","abstract_html":"To effectively build a regression model with a large number of covariates is no easy task. We consider using dimension reduction before building a parametric or spline model. The dimension reduction procedure is based on a canonical correlation analysis on the predictor variables and a spline basis generated for the response variable. One important question in dimension reduction is to decide on the number of effective dimensions needed. We study four tests of dimensionality: a chi-square test, a Wald-type test on eigenvalues, a modified Wald-type test, and a matrix rank test. These tests are motivated from different aspects of the problem and have their own strength and weakness. We discuss and compare these tests both theoretically and through Monte Carlo simulations, based on which specific recommendations for determining dimensionality are made. Additive regression splines are first fitted to the data in the space of reduced dimensionality. A Tukey-type test of additivity is proposed and compared with Rao's score test. 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