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.

在過去研究犯罪經濟學的理論文獻上，失業率對各犯罪類型的影響為正向關係，但在實証文獻上的研究發現，卻有愈來愈多的証據支持此二個變數間的負向或無關係。為了解釋上述正向與負向間相反的矛盾關係，本篇論文嘗試利用兩種模型(非參數與非參數模型)與兩種效果(機會效果與動機效果)來解釋此二變數間的關係，此亦是本論文主要貢獻。其中機會效果是用以解釋失業率與犯罪間的負向關係，動機效果則用以解釋正向關係。在非參數模型中，利用失業率為景氣循環的代理變數，發現失業率與竊盜間存在正向關係，此與大多實証研究相符；失業率則和妨害風化與殺人犯罪間呈現負向相關；失業率與傷害罪間則沒有明顯正負關係。研究亦顯示，不同的犯罪類型在不同的參數模型下，統計的顯著性亦有不同，而在不同年齡層(青少年與成年人)的犯罪模型則更與理論模型結論相符。 / According to the theoretical literature on criminal economics, unemployment rate tends to be positively correlated to all types of crimes. However, more and more empirical evidence suggests otherwise. In order to clarify the relationship, this study exploits both nonparametric and parametric models and considers two effects, including opportunity and motivation effects. The presence of the opportunity effect leads to be a negative correlation between unemployment rate and crimes, while the presence of the motivation effect gives a positive correlation. Under nonparametric model where unemployment rate is used as a proxy for business cycles, we only found that there is positive correlation between unemployment rate and robbery, while obscenity and homicide are found to be negatively correlated with unemployment rate. This is in line with most empirical studies. Little correlation evidence is found for unemployment and other types of crimes. Under parametric model, the study indicates that the statistical significance differs in models, and depends on crime variable used. We found more consistent results with theoretic models for the age groups (teenagers and adults).

在迴歸分析中，若變數間具有非線性 (nonlinear) 的關係時，B-Spline線性迴歸是以無母數的方式建立模型。B-Spline函數為具有節點(knots)的分段多項式，選取合適節點的位置對B-Spline函數的估計有重要的影響，在希望得到B-Spline較好的估計量的同時，我們也想要只用少數的節點就達成想要的成效，於是Huang (2013) 提出了一種選擇節點的方式APLS (Adaptive penalized least squares)，在本文中，我們以此方法進行一些更一般化的設定，並在不同的設定之下，判斷是否有較好的估計效果，且已修正後的方法與基於BIC (Bayesian information criterion)的節點估計方式進行比較，在本文中我們將一般化設定的APLS法稱為GAPLS，並且經由模擬結果我們發現此兩種以B-Spline進行迴歸函數近似的方法其近似效果都很不錯，只是節點的個數略有不同，所以若是對節點選取的個數有嚴格要求要取較少的節點的話，我們建議使用基於BIC的節點估計方式，除此之外GAPLS法也是不錯的選擇。 / In regression analysis, if the relationship between the response variable and the explanatory variables is nonlinear, B-splines can be used to model the nonlinear relationship. Knot selection is crucial in B-spline regression. Huang (2013) propose a method for adaptive estimation, where knots are selected based on penalized least squares. This method is abbreviated as APLS (adaptive penalized least squares) in this thesis. In this thesis, a more general version of APLS is proposed, which is abbreviated as GAPLS (generalized APLS). Simulation studies are carried out to compare the estimation performance between GAPLS and a knot selection method based on BIC (Bayesian information criterion). The simulation results show that both methods perform well and fewer knots are selected using the BIC approach than using GAPLS.