迴歸分析中，若變數間有非線性(nonlinear)的關係，此時我們可以用B-spline線性迴歸，一種無母數的方法，建立模型。Penalized spline是B-spline方法的一種改良，其想法是增加一懲罰項，避免估計函數時出現過度配適的問題。本文中，考慮三種方法:(a) Marginal Mixed Model approach, (b) Conditional Mixed Model approach, (c) 貝氏方法建立信賴帶，其中，我們對第一二種方法內的估計式作了一點調整，另外，懲罰項中的平滑參數也是我們考慮的問題。我們發現平滑參數確實會影響信賴帶，所以我們使用cross-validation來選取平滑參數。在調整的cross-validation下，Marginal Mixed Model的信賴帶估計不平滑的函數效果較好，Conditional Mixed Model的信賴帶估計平滑函數的效果較好，貝氏的信賴帶估計函數效果較差。 / In regression analysis, we can use B-spline to estimate regression function nonparametrically when the regression function is nonlinear. Penalized splines have been proposed to improve the performance of B-splines by including a penalty term to prevent over-fitting. In this article, we compare confidence bands constructed by three estimation methods: (a) Marginal Mixed Model approach, (b) Conditional Mixed Model approach, and (c) Bayesian approach. We modify the first two methods slightly. In addition, the selection of smoothing parameter of penalization is considered. We found that the smoothing parameter affects confidence bands a lot, so we use cross-validation to choose the smoothing parameter. Finally, based on the restricted cross-validation, Marginal Mixed Model performs better for less smooth regression functions, Conditional Mixed Model performs better for smooth regression functions and Bayesian approach performs badly.

資本資產定價模型已被廣泛使用於分析股票風險與要求報酬率之間的關係。然而，個別股票風險Beta是否足以解釋其報酬，也受到愈來愈多的質疑。Fama和French在1993年提出額外兩個因子來解釋股票報酬。我們將應用資本資產定價模型和三因子模型來分析1963年7月至2002年12月之美國的三大股票交易所上市公司。藉由一次改變分析過程中的一部分，以觀察參數估計值是否穩定。結果發現Beta_HML總是顯著且最為穩定，而Beta_SMB並不顯著。Beta經常顯著，但變動情況較大。另外，我們將考慮個別股票本身的變異，亦即將隨機效果納入考量。 / The Capital Asset Pricing Model (CAPM) has been widely used to analyze the relationship between risk and required rate of return on a stock, while it is doubted that individual stock's risk Beta has enough explanatory power for it's returns. Fama and French (1993) proposed two more factors to help explaining stock returns. We use the CAPM and the three-factor model to analyze listed companys in American stock exchanges, during the period from July 1963 to December 2002. We change part of the analyzing process a time to see if the estimates of the parameters are stable. The risk-premium Beta_HML is always significant and it performs most stable, while another risk-premium Beta_SMB is never significant. Beta is usually significant but it varies. Furthermore, we take within-stock variation into account, so random effects are considered.