分量迴歸(Quantile regression, QR)透過中位數來描述變數間的關聯性，其後延伸到任意不同的分量值，皆可以使用分量迴歸來建立模型描述變數間的關聯性，因此分量迴歸可以應用到各式各樣的資料和領域上。Zou and Yuan (2008)提出在不同分量下，彼此之間除了截距項以外的迴歸係數假設皆相同，我們可以將所有分量下的損失函數加總作參數估計，這個新的分量迴歸估計方法稱為Composite quantile regression(CQR)。由於使用QR或CQR參數估計前需先檢定資料傾向哪種估計方法的前提假設，需要二階段的動作才能得到答案，況且不是CQR就一定得使用QR來做參數估計的話，未免有些沒效率；因此在考慮同時進行選定模型和參數估計的想法下，我們嘗試在不同分量下，將所有分量迴歸的損失函數加總後，再以加入設限函數的方式，建構一個新的分量迴歸係數估計的方法，稱之為設限分量迴歸(Penalized quantile regression)，使得在參數估計的過程中可以同時進行選定模型的動作，其中PQR估計量的漸近分佈和QR的漸近分佈會相同；而在模擬分析中也可以發現，在PQR、QR、CQR的參數估計量比較之下，PQR皆可以有不錯的表現。

Quantile regression (QR) describes the relationship between the response variable and the exploratory variables through some specific quantiles, which has been applied to a wide range of data and different fields. Under the linear model assumption, Zou and Yuan (2008) proposed the composite quantile regression (CQR) to incorporate several quantiles at a time in the estimation function. In theory, CQR has better estimation precision when the linear model assumption holds, but it is not adequate and biased when the assumption is wrong. Without the linear model assumption, this thesis suggest a penalized quantile regression (PQR) method which implements either QR or CQR according to the empirical data property, by including a specific grouped lasso regularization term on the regression parameters in the estimation function. Simulation results show that PQR has good estimation performance over the QR and CQR under various situations.

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