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Local shrinkage rules, Lévy processes and regularized regression. (English) Zbl 1411.62209

Summary: We use Lévy processes to generate joint prior distributions, and therefore penalty functions, for a location parameter \(\beta=(\beta_1,\ldots,\beta_p)\) as \(p\) grows large. This generalizes the class of local-global shrinkage rules based on scale mixtures of normals, illuminates new connections between disparate methods and leads to new results for computing posterior means and modes under a wide class of priors. We extend this framework to large-scale regularized regression problems where \(p>n\), and we provide comparisons with other methodologies.

MSC:

62J07 Ridge regression; shrinkage estimators (Lasso)
62H25 Factor analysis and principal components; correspondence analysis
62F12 Asymptotic properties of parametric estimators
60G51 Processes with independent increments; Lévy processes
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