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Weak Convergence of the Regularization Path in Penalized M-Estimation




We consider a function defined as the pointwise minimization of a doubly index random process. We are interested in the weak convergence of the minimizer in the space of bounded functions. Such convergence results can be applied in the context of penalized M-estimation, that is, when the random process to minimize is expressed as a goodness-of-fit term plus a penalty term multiplied by a penalty weight. This weight is called the "regularization parameter" and the minimizing function the "regularization path". The regularization path can be seen as a collection of estimators indexed by the regularization parameter. We obtain a consistency result and a central limit theorem for the regularization path in a functional sense. Various examples are provided, including the ℓ-super-1-regularization path for general linear models, the ℓ-super-1- or ℓ-super-2-regularization path of the least absolute deviation regression and the Akaike information criterion. Copyright (c) 2010 Board of the Foundation of the Scandinavian Journal of Statistics.

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  • Jean-Francois Germain & Francois Roueff, 2010. "Weak Convergence of the Regularization Path in Penalized M-Estimation," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(3), pages 477-495.
  • Handle: RePEc:bla:scjsta:v:37:y:2010:i:3:p:477-495

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    References listed on IDEAS

    1. Pollard, David, 1985. "New Ways to Prove Central Limit Theorems," Econometric Theory, Cambridge University Press, vol. 1(03), pages 295-313, December.
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    Cited by:

    1. Ban Zheng & Eric Moulines & Fr'ed'eric Abergel, 2012. "Price Jump Prediction in Limit Order Book," Papers 1204.1381,
    2. repec:hal:wpaper:hal-00684716 is not listed on IDEAS
    3. Ban Zheng & Eric Moulines & Frédéric Abergel, 2013. "Price jump prediction in a limit order book," Post-Print hal-00684716, HAL.

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