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We propose an adaptive approach based on semiparametric models, which we call the (conditional) minimum average variance estimation (MAVE) method, within quite a general setting. The MAVE method has the following advantages. Most existing methods must undersmooth the nonparametric link function estimator to achieve a faster rate of consistency for the estimator of the parameters (than for that of the nonparametric function). In contrast, a faster consistency rate can be achieved by the MAVE method even without undersmoothing the nonparametric link function estimator. The MAVE method is applicable to a wide range of models, with fewer restrictions on the distribution of the covariates, to the extent that even time series can be included. Because of the faster rate of consistency for the parameter estimators, it is possible for us to estimate the dimension of the space consistently. The relationship of the MAVE method with other methods is also investigated. In particular, a simple outer product gradient estimator is proposed as an initial estimator. In addition to theoretical results, we demonstrate the efficacy of the MAVE method for high dimensional data sets through simulation. Two real data sets are analysed by using the MAVE approach.<\/jats:p>","DOI":"10.1111\/1467-9868.03411","type":"journal-article","created":{"date-parts":[[2003,3,12]],"date-time":"2003-03-12T14:37:22Z","timestamp":1047479842000},"page":"363-410","source":"Crossref","is-referenced-by-count":653,"title":["An Adaptive Estimation of Dimension Reduction Space"],"prefix":"10.1093","volume":"64","author":[{"given":"Yingcun","family":"Xia","sequence":"first","affiliation":[{"name":"University of Cambridge , UK"},{"name":"Jinan University , People\u2019s Republic of China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Howell","family":"Tong","sequence":"additional","affiliation":[{"name":"University of Hong Kong , People\u2019s Republic of China"},{"name":"London School of Economics and Political Science , UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"W. K.","family":"Li","sequence":"additional","affiliation":[{"name":"University of Hong Kong , People\u2019s Republic of China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Li-Xing","family":"Zhu","sequence":"additional","affiliation":[{"name":"University of Hong Kong and Chinese Academy of Sciences , Beijing , People\u2019s Republic of China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2002,8,12]]},"reference":[{"key":"2023040109112480800_","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1093\/biomet\/77.4.669","article-title":"Identification of nonlinear time series: first order characterization and order determination","volume":"77","author":"Auestad","year":"1990","journal-title":"Biometrika"},{"key":"2023040109112480800_","volume-title":"Classification and Regression Trees","author":"Breiman","year":"1984"},{"key":"2023040109112480800_","doi-asserted-by":"crossref","first-page":"941","DOI":"10.1080\/01621459.2000.10474284","article-title":"Functional-coefficient regression models for nonlinear time series","volume":"95","author":"Cai","year":"2000","journal-title":"J. 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