在傳統上，估計與選模大多分開來討論。Tibshrini (1996) 提出Lasso估計式，可以同時完成參數的估計與模型的選取，Lasso的概念是在概似函數上加入一個限制式，迫使不重要的迴歸係數等於0，參數估計等同於模型選取，同時選出的模型不容易受資料些微的變動所影響，但Lasso也因加入限制式的關係，其估計量不具備一致性，於是Zou(2006) 提出 Adaptive Lasso 來修正Lasso，使其估計滿足Oracle性質。當資料有類別變數時，Yuan 和 Lin(2004)提出Group Lasso修正Lasso對於類別變數的選取，其group變數(啞變數組成)應要同進同出，也就是不能只挑選group變數內的部分變數，應要全部選入或是剔除group變數，因為Group Lasso是Lasso的一個推廣，所以參數估計仍然不具備一致性。；Wang 和 Leng (2006)提出Adaptive Group Lasso 來修正 Group Lasso使其滿足Oracle性質；本研究將利用Adaptive Group Lasso在多變量線性模性的模型選取與參數估計，這裡的Adaptive Group Lasso的定義不同於先前將X矩陣的每行視做一群，而是將參數矩陣每列視為一群，目的是想了解那些變數對Y有影響。

In traditional statistical method, estimation and variable selection are almost discussed separately. LASSO (Tibshirani, 1996) is a new method for estimation in linear model, it can estimate parameters and variable selection simultaneously. But Lasso is inconsistent for variable selection, Adaptive Lasso (Zou 2006) overcomes these problems and enjoys the oracle properties. In linear regression when categorical predictors (factors) are present, the Lasso solution only selects individual dummy variables instead of whole factors. The group Lasso(Yuan and Lin 2006) overcomes these problems. Group lasso is a natural extension of lasso and selects variable in a grouped manner, group lasso suffers from estimation inefficiency and selection inconsistency. Adaptive Group Lasso (Wang and Leng 2006) show it’s estimator can be as efficient as oracle. We propose the adaptive group lasso for multivariate linear regression. In our study, the definition of grouped variable is different with the definition defined by formed study, which is regard one column of model matrix as a group. We consider one row of parametric matrix as one group for finding the significant variable on Y.

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