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。

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.

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

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