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Abstract:
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Variable selection is an important topic in linear regression analysis and attracts a lot of research in this era of big data. It is fundamental to high-dimensional statistical modeling, including nonparametric regression. Some classic techniques include stepwise deletion and subset selection. However, these procedures ignore stochastic errors inherited in the stages of variable selections and the resulting subset suffers from the lack of stability. Penalized least squares provide new approaches to the variable selection problems. The LASSO which imposes an L1-penalty on the regression coefficients and the elastic net which combines an L1 and an L2 penalties are popular members of the penalized least squares. In this research, we develop penalized regressions of a penalty function ||?||_?^? and prove that the LASSO and elastic net are special cases of our function. The structure and properties of ||?||_?^? penalty are studied and the corresponding algorithms are developed. Simulation studies and real-data support the advantageous performance of the proposed methods. Some other related topics may also be discussed, for example, multiple independent or correlated responses in linear
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