IDEAS home Printed from
   My bibliography  Save this paper

Estimating linear functionals of a sparse family of Poisson means Price Discrimination


  • Olivier Collier

    (Modal'X; Université Paris-Nanterre;CREST; ENSAE)

  • Arnak Dalalyan

    (Modal'X; Université Paris-Nanterre;CREST; ENSAE)


Assume that we observe a sample of size n composed of p-dimensional signals, each signal having independent entries drawn from a scaled Poisson distribution with an unknown intensity. We are interested in estimating the sum of the n unknown intensity vectors, under the assumption that most of them coincide with a given "background" signal. The number s of p-dimensional signals different from the background signal plays the role of sparsity and the goal is to leverage this sparsity assumption in order to improve the quality of estimation as compared to the naive estimator that computes the sum of the observed signals. We first introduce the group hard thresholding estimator and analyze its mean squared error measured by the squared Euclidean norm. We establish a nonasymptotic upper bound showing that the risk is at most of the order of thetha^2(sp + s^2 * sqrt(p)) log^3/2(np). We then establish lower bounds on the minimax risk over a properly defined class of collections of s-sparse signals. These lower bounds match with the upper bound, up to logarithmic terms, when the dimension p is fixed or of larger order than s^2. In the case where the dimension p increases but remains of smaller order than s^2, our results show a gap between the lower and the upper bounds, which can be up to order sqrt(p).

Suggested Citation

  • Olivier Collier & Arnak Dalalyan, 2017. "Estimating linear functionals of a sparse family of Poisson means Price Discrimination," Working Papers 2017-19, Center for Research in Economics and Statistics.
  • Handle: RePEc:crs:wpaper:2017-19

    Download full text from publisher

    File URL:
    File Function: CREST working paper version
    Download Restriction: no

    References listed on IDEAS

    1. Kutoyants, Yu. A. & Liese, F., 1998. "Estimation of linear functionals of Poisson processes," Statistics & Probability Letters, Elsevier, vol. 40(1), pages 43-55, September.
    2. Laetitia Comminges & Arnak Dalalyan, 2012. "Minimax Testing of a Composite null Hypothesis Defined via a Quadratic Functional in the Model of regression," Working Papers 2012-19, Center for Research in Economics and Statistics.
    3. Olivier Collier & Arnak S, Dalalyan, 2013. "Curve registration by Nonparametric goodness-of-fit Testing," Working Papers 2013-33, Center for Research in Economics and Statistics.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Olivier Collier & Arnak S. Dalalyan, 2018. "Estimating linear functionals of a sparse family of Poisson means," Statistical Inference for Stochastic Processes, Springer, vol. 21(2), pages 331-344, July.
    2. del Barrio, Eustasio & Gordaliza, Paula & Lescornel, Hélène & Loubes, Jean-Michel, 2019. "Central limit theorem and bootstrap procedure for Wasserstein’s variations with an application to structural relationships between distributions," Journal of Multivariate Analysis, Elsevier, vol. 169(C), pages 341-362.
    3. Holger Dette & Subhra Sankar Dhar & Weichi Wu, 2021. "Identifying shifts between two regression curves," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 73(5), pages 855-889, October.

    More about this item


    Nonasymptotic minimax estimation; linear functional; group-sparsity; thresholding; Poisson processes;
    All these keywords.

    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:crs:wpaper:2017-19. See general information about how to correct material in RePEc.

    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 bibliographic 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.

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

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

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.