IDEAS home Printed from https://ideas.repec.org/a/bla/jorssb/v70y2008i3p461-493.html
   My bibliography  Save this article

Proportion of non‐zero normal means: universal oracle equivalences and uniformly consistent estimators

Author

Listed:
  • Jiashun Jin

Abstract

Summary. Since James and Stein's seminal work, the problem of estimating n normal means has received plenty of enthusiasm in the statistics community. Recently, driven by the fast expansion of the field of large‐scale multiple testing, there has been a resurgence of research interest in the n normal means problem. The new interest, however, is more or less concentrated on testing n normal means: to determine simultaneously which means are 0 and which are not. In this setting, the proportion of the non‐zero means plays a key role. Motivated by examples in genomics and astronomy, we are particularly interested in estimating the proportion of non‐zero means, i.e. given n independent normal random variables with individual means Xj∼N(μj,1), j=1,…,n, to estimate the proportion ɛn=(1/n) #{j:μj /= 0}. We propose a general approach to construct the universal oracle equivalence of the proportion. The construction is based on the underlying characteristic function. The oracle equivalence reduces the problem of estimating the proportion to the problem of estimating the oracle, which is relatively easier to handle. In fact, the oracle equivalence naturally yields a family of estimators for the proportion, which are consistent under mild conditions, uniformly across a wide class of parameters. The approach compares favourably with recent works by Meinshausen and Rice, and Genovese and Wasserman. In particular, the consistency is proved for an unprecedentedly broad class of situations; the class is almost the largest that can be hoped for without further constraints on the model. We also discuss various extensions of the approach, report results on simulation experiments and make connections between the approach and several recent procedures in large‐scale multiple testing, including the false discovery rate approach and the local false discovery rate approach.

Suggested Citation

  • Jiashun Jin, 2008. "Proportion of non‐zero normal means: universal oracle equivalences and uniformly consistent estimators," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(3), pages 461-493, July.
  • Handle: RePEc:bla:jorssb:v:70:y:2008:i:3:p:461-493
    DOI: 10.1111/j.1467-9868.2007.00645.x
    as

    Download full text from publisher

    File URL: https://doi.org/10.1111/j.1467-9868.2007.00645.x
    Download Restriction: no

    File URL: https://libkey.io/10.1111/j.1467-9868.2007.00645.x?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    Citations

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


    Cited by:

    1. Jiaying Gu & Roger Koenker, 2016. "On a Problem of Robbins," International Statistical Review, International Statistical Institute, vol. 84(2), pages 224-244, August.
    2. Helmut Finner & Veronika Gontscharuk, 2009. "Controlling the familywise error rate with plug‐in estimator for the proportion of true null hypotheses," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(5), pages 1031-1048, November.
    3. Jessie Jeng, X., 2016. "Detecting weak signals in high dimensions," Journal of Multivariate Analysis, Elsevier, vol. 147(C), pages 234-246.
    4. Chen, Xiongzhi, 2019. "Uniformly consistently estimating the proportion of false null hypotheses via Lebesgue–Stieltjes integral equations," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 724-744.
    5. T. Tony Cai & Wenguang Sun, 2017. "Optimal screening and discovery of sparse signals with applications to multistage high throughput studies," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(1), pages 197-223, January.
    6. Li Wang, 2019. "Weighted multiple testing procedure for grouped hypotheses with k-FWER control," Computational Statistics, Springer, vol. 34(2), pages 885-909, June.
    7. Rohit Kumar Patra & Bodhisattva Sen, 2016. "Estimation of a two-component mixture model with applications to multiple testing," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(4), pages 869-893, September.
    8. Cipolli III, William & Hanson, Timothy & McLain, Alexander C., 2016. "Bayesian nonparametric multiple testing," Computational Statistics & Data Analysis, Elsevier, vol. 101(C), pages 64-79.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    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:bla:jorssb:v:70:y:2008:i:3:p:461-493. 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.

    We have no bibliographic references for this item. You can help adding them by using 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: Wiley Content Delivery (email available below). General contact details of provider: https://edirc.repec.org/data/rssssea.html .

    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.