Mathematics – Statistics Theory
Scientific paper
2008-05-21
IMS Collections 2008, Vol. 3, 122-137
Mathematics
Statistics Theory
Published in at http://dx.doi.org/10.1214/074921708000000101 the IMS Collections (http://www.imstat.org/publications/imscollec
Scientific paper
10.1214/074921708000000101
In this article we investigate consistency of selection in regression models via the popular Lasso method. Here we depart from the traditional linear regression assumption and consider approximations of the regression function $f$ with elements of a given dictionary of $M$ functions. The target for consistency is the index set of those functions from this dictionary that realize the most parsimonious approximation to $f$ among all linear combinations belonging to an $L_2$ ball centered at $f$ and of radius $r_{n,M}^2$. In this framework we show that a consistent estimate of this index set can be derived via $\ell_1$ penalized least squares, with a data dependent penalty and with tuning sequence $r_{n,M}>\sqrt{\log(Mn)/n}$, where $n$ is the sample size. Our results hold for any $1\leq M\leq n^{\gamma}$, for any $\gamma>0$.
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