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Removing the singularity of a penalty via thresholding function matching
Abstract:By introducing the idea of thresholding function matching, it is illustrated that both bridge penalty and log penalty can be transformed so as to circumvent certain difficulties in numerical computation and the definition of local minimality. The fact that both bridge penalty and log penalty have derivatives going to infinity at zero. This hinders their applications in statistics although it is reported in the literature that they allow recovery of sparse structure in the data under some conditions. It is illustrated in the simulation studies that in the variable selection problems, penalized likelihood estimation based on the transformed penalty obtained by the proposed thresholding function matching method outperform those based on many other state-of-art penalties, particularly when the covariates are strongly correlated. The one-to-one correspondence between the transformed penalties and their thresholding functions are also established.
Keywords:Penalized regression problem  Penalty function  Reconstructable penalty  Thresholding function  Transformable penalty
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