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Such RSS-valued data arise frequently in medical imaging, surface modelling and computer vision, among many other fields. We develop an intrinsic regression model solely based on an intrinsic conditional moment assumption, avoiding specifying any parametric distribution in RSS. We propose various link functions to map from the Euclidean space of multiple covariates to the RSS of responses. We develop a two-stage procedure to calculate the parameter estimates and determine their asymptotic distributions. We construct the Wald and geodesic test statistics to test hypotheses of unknown parameters. We systematically investigate the geometric invariant property of these estimates and test statistics. Simulation studies and a real data analysis are used to evaluate the finite sample properties of our methods.<\/jats:p>","DOI":"10.1111\/rssb.12169","type":"journal-article","created":{"date-parts":[[2016,3,21]],"date-time":"2016-03-21T00:42:49Z","timestamp":1458520969000},"page":"463-482","source":"Crossref","is-referenced-by-count":71,"title":["Regression Models on Riemannian Symmetric Spaces"],"prefix":"10.1093","volume":"79","author":[{"given":"Emil","family":"Cornea","sequence":"first","affiliation":[{"name":"University of North Carolina at Chapel Hill \u00a0 USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hongtu","family":"Zhu","sequence":"additional","affiliation":[{"name":"University of North Carolina at Chapel Hill \u00a0 USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Peter","family":"Kim","sequence":"additional","affiliation":[{"name":"University of Guelph \u00a0 Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Joseph G.","family":"Ibrahim","sequence":"additional","affiliation":[{"name":"University of North Carolina at Chapel Hill \u00a0 USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2016,3,20]]},"reference":[{"key":"2023021709342732100_rssb12169-cit-0001","doi-asserted-by":"crossref","first-page":"851","DOI":"10.1093\/biomet\/asq044","article-title":"Nonparametric Bayesian density estimation on manifolds with applications to planar shapes","volume":"97","author":"Bhattacharya","year":"2010","journal-title":"Biometrika"},{"key":"2023021709342732100_rssb12169-cit-0002","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jmva.2012.02.020","article-title":"Nonparametric Bayes classification and hypothesis testing on manifolds","volume":"111","author":"Bhattacharya","year":"2012","journal-title":"J. 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