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Weak convergence of the regularization path in penalized M-estimation

JEAN‐FRANCOIS Germain
François Roueff
Abstract

We consider an estimator \hbbetan(t)\hbbeta_n(t) defined as the element \bphi\bPhi\bphi\in\bPhi minimizing a contrast process \pencontrast(\bphi,t)\pencontrast(\bphi, t) for each t. We give some general results for deriving the weak convergence of n(\hbbetan\bbeta)\sqrt{n}(\hbbeta_n-\bbeta) in the space of bounded functions, where, for each t, \bbeta(t)\bbeta(t) is the \bphi\bPhi\bphi\in\bPhi minimizing the limit of \pencontrast(\bphi,t)\pencontrast(\bphi, t) as nn\to\infty. These results are applied in the context of penalized M-estimation, that is, when \pencontrast(\bphi,t)=Mn(\bphi)+tJn(\bphi)\pencontrast(\bphi, t)=M_n(\bphi)+ t J_n(\bphi), where MnM_n is a usual contrast process and JnJ_n a penalty such as the 1\ell^1 norm or the squared 2\ell^2 norm. The function \hbbetan\hbbeta_n is then called a \emph{regularization path}. For instance we show that the central limit theorem established for the lasso estimator in Knight and Fu (2000) continues to hold in a functional sense for the regularization path. Other examples include various possible contrast processes for MnM_n such as those considered in Pollard (1985).

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