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 系統識別號 U0002-2007201610335300 中文論文名稱 單指數模型適合度檢定之模擬研究 英文論文名稱 A simulation study of the goodness-of-fit test for single-index model 校院名稱 淡江大學 系所名稱(中) 數學學系碩士班 系所名稱(英) Department of Mathematics 學年度 104 學期 2 出版年 105 研究生中文姓名 陳孟廷 研究生英文姓名 Mong-Ting Cheng 學號 603190082 學位類別 碩士 語文別 英文 口試日期 2016-06-17 論文頁數 28頁 口試委員 指導教授-黃逸輝委員-温啟仲委員-黃文瀚 中文關鍵字 正交  格蘭姆-施密特法  核密度估計 英文關鍵字 Inner product  Gram-Schmidt process  Kernel density estimation 學科別分類 學科別＞自然科學＞數學 中文摘要 本論文主要是利用Huang & Wen (2016)所發展的半母數模型適合度檢定套用在單指數模型之模擬研究。如果模型假設是正確的，則殘差理應會跟自變數X所張出的空間垂直，反之，則殘差理應會跟自變數X所張出的空間不垂直。因為有這個簡單的幾何關係， Huang (2016)利用如果垂直則內積是零的觀念，發展出了模型的適合度檢定方法，然而這個方法是否可以套用在單指數模型及表現，我們仍然未知，所以在本論文進行幾個簡單的模擬研究來檢視這個方法並探討和研究寬度h、參數β ,變異數σ2 及樣本數對檢定力的影響。 英文摘要 Single-Index models relax restrictive part of usual assumptions on parametric models. They are becoming popular in these years and are used in a various field statistic of economic, financial sector, biostatistics, medical science fields. In this paper, we study goodness-of-fit testing for Single-Index models. The semi-parametric approach make the complexity in the assessing of validness of the model. So, we adopted Huang & Wen (2016) approach. If the model is correctly specified, then the residual of the fitted model is orthogonal to functions of covariates. On the other hand, if the model assumption is incorrect, the residual will not be orthogonal to covariates. With this geometric relation, Huang (2016) develops a goodness-of-fit test based on whether certain inner products have zero means. In our thesis, we assess the performs of such projection approach by various simulation study. This is a novel approach and its performance is still unclear. We conduct a few simulation studies to asses the performance of the approach, and investigate the impacts of the bandwidth h selection, β ,σ2 and sample size on the power of the test. 論文目次 Contents 1. Introduction 1 2. The constructing testing statistic 2 2.1 Model Estimation 2 2.2 Model Checking 3 3. Simulation study 9 4. Conclusion 26 Appendix 27 References 27 參考文獻 Carroll, R. J., & Fan, J., & Gijbels, I., & Wand, M. P. (1997). Generalized Partially Linear Single-Index Models.J. Amer. Statis. Assn.92,477-489. Dong, C., & Gao, J., & Tjostheim, D. (2016). Estimation for Single-Index and Partially Linear Single-Index Integrated Models.Ann. Statis.44,425-453 Guidoum, A., C. (2015). Kernel Estimator and Bandwidth Selection for Density and its Derivatives. Hardle, W., & Hall, P., & Ichimura, H. (1993). Optimal Smoothing In Single-Index models.Ann. Statis.21,157-178 Huang, Y. H., & Wen, C. C. (2016). A goodness-of-fi t test for parametric mean functions.Preparing. Huang, Y. H.(2016). The projection approach for goodness-of-fi t test on single index model.Preparing. Liang, H., & Lui, X., & Li, R., & Tsai, C. H. (2010). Estimation and Testing for Partially Linear Single-Index Models.Ann. Statist.38,3811-3836. Nadaraya, E. A. (1964). On Estimating Regression.Theory Probab. Appl.9,141-142. Stute, W., & Zhu, L. X. (2005). Nonparametric Checks for Single-Index Models.Ann. Statist.33,1048-1083. Wang, J. L., & Xue, L., & Zhu, L., & Chong, Y. S. (2010). Estimation for a Partial-Linear Single-Index model.Ann. Statis.38,246-274 Xia, Y., & Li, W. K., & Tong, H., & Zhang, Z. (2004). A Goodness-of-Fit Test for Single-Index Models.Statist. Sinica.14,1-39. 論文使用權限 同意紙本無償授權給館內讀者為學術之目的重製使用，於2016-07-21公開。同意授權瀏覽/列印電子全文服務，於2016-07-21起公開。

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