IDEAS home Printed from
   My bibliography  Save this paper

Adaptive Minimax Estimation over Sparse l q-Hulls


  • Zhan Wang


  • Sandra Paterlini


  • Fuchang Gao


  • Yuhong Tang



Given a dictionary of $M_n$ initial estimates of the unknown true regression function, we aim to construct linearly aggregated estimators that target the best performance among all the linear combinations under a sparse $q$-norm ($0 \leq q \leq 1$) constraint on the linear coefficients. Besides identifying the optimal rates of aggregation for these $\ell_q$-aggregation problems, our multi-directional (or universal) aggregation strategies by model mixing or model selection achieve the optimal rates simultaneously over the full range of $0\leq q \leq 1$ for general $M_n$ and upper bound $t_n$ of the $q$-norm. Both random and fixed designs, with known or unknown error variance, are handled, and the $\ell_q$-aggregations examined in this work cover major types of aggregation problems previously studied in the literature. Consequences on minimax-rate adaptive regression under $\ell_q$-constrained true coefficients ($0 \leq q \leq 1$) are also provided. Our results show that the minimax rate of $\ell_q$-aggregation ($0 \leq q \leq 1$) is basically determined by an effective model size, which is a sparsity index that depends on $q$, $t_n$, $M_n$, and the sample size $n$ in an easily interpretable way based on a classical model selection theory that deals with a large number of models. In addition, in the fixed design case, the model selection approach is seen to yield optimal rates of convergence not only in expectation but also with exponential decay of deviation probability. In contrast, the model mixing approach can have leading constant one in front of the target risk in the oracle inequality while not offering optimality in deviation probability.

Suggested Citation

  • Zhan Wang & Sandra Paterlini & Fuchang Gao & Yuhong Tang, 2012. "Adaptive Minimax Estimation over Sparse l q-Hulls," Department of Economics 0681, University of Modena and Reggio E., Faculty of Economics "Marco Biagi".
  • Handle: RePEc:mod:depeco:0681

    Download full text from publisher

    File URL:
    Download Restriction: no

    Other versions of this item:

    References listed on IDEAS

    1. Zhan Wang & Sandra Paterlini & Fuchang Gao & Yuhong Yang, 2012. "Adaptive Minimax Estimation over Sparse lq-Hulls," Center for Economic Research (RECent) 078, University of Modena and Reggio E., Dept. of Economics "Marco Biagi".
    Full references (including those not matched with items on IDEAS)


    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.

    Cited by:

    1. Zhan Wang & Sandra Paterlini & Fuchang Gao & Yuhong Tang, 2012. "Adaptive Minimax Estimation over Sparse l q-Hulls," Department of Economics 0681, University of Modena and Reggio E., Faculty of Economics "Marco Biagi".

    More about this item


    minimax risk; adaptive estimation; linear combining; aggregation; model mixing; model selection;

    NEP fields

    This paper has been announced in the following NEP Reports:


    Access and download statistics


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:mod:depeco:0681. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Sara Colombini). General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.