IDEAS home Printed from https://ideas.repec.org/a/spr/metcap/v14y2012i3d10.1007_s11009-011-9233-z.html
   My bibliography  Save this article

Inference on High-Dimensional Mean Vectors with Fewer Observations Than the Dimension

Author

Listed:
  • Kazuyoshi Yata

    (University of Tsukuba)

  • Makoto Aoshima

    (University of Tsukuba)

Abstract

We focus on inference about high-dimensional mean vectors when the sample size is much fewer than the dimension. Such data situation occurs in many areas of modern science such as genetic microarrays, medical imaging, text recognition, finance, chemometrics, and so on. First, we give a given-radius confidence region for mean vectors. This inference can be utilized as a variable selection of high-dimensional data. Next, we give a given-width confidence interval for squared norm of mean vectors. This inference can be utilized in a classification procedure of high-dimensional data. In order to assure a prespecified coverage probability, we propose a two-stage estimation methodology and determine the required sample size for each inference. Finally, we demonstrate how the new methodologies perform by using a microarray data set.

Suggested Citation

  • Kazuyoshi Yata & Makoto Aoshima, 2012. "Inference on High-Dimensional Mean Vectors with Fewer Observations Than the Dimension," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 459-476, September.
    • Handle: RePEc:spr:metcap:v:14:y:2012:i:3:d:10.1007_s11009-011-9233-z
    DOI: 10.1007/s11009-011-9233-z
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11009-011-9233-z
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11009-011-9233-z?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
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Peter Hall & J. S. Marron & Amnon Neeman, 2005. "Geometric representation of high dimension, low sample size data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(3), pages 427-444, June.
    2. N. Mukhopadhyay & T. Solanky, 1997. "Estimation after sequential selection and ranking," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 45(1), pages 95-106, January.
    3. Makoto Aoshima & Kazuyoshi Yata, 2010. "Asymptotic second-order consistency for two-stage estimation methodologies and its applications," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 62(3), pages 571-600, June.
    4. Aoshima, Makoto & Mukhopadhyay, Nitis, 1998. "Fixed-Width Simultaneous Confidence Intervals for Multinormal Means in Several Intraclass Correlation Models," Journal of Multivariate Analysis, Elsevier, vol. 66(1), pages 46-63, July.
    5. Jeongyoun Ahn & J. S. Marron & Keith M. Muller & Yueh-Yun Chi, 2007. "The high-dimension, low-sample-size geometric representation holds under mild conditions," Biometrika, Biometrika Trust, vol. 94(3), pages 760-766.
    Full references (including those not matched with items on IDEAS)

    Citations

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


    Cited by:

    1. Jun Li, 2018. "Asymptotic normality of interpoint distances for high-dimensional data with applications to the two-sample problem," Biometrika, Biometrika Trust, vol. 105(3), pages 529-546.
    2. Makoto Aoshima & Kazuyoshi Yata, 2015. "Asymptotic Normality for Inference on Multisample, High-Dimensional Mean Vectors Under Mild Conditions," Methodology and Computing in Applied Probability, Springer, vol. 17(2), pages 419-439, June.

    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. Nitis Mukhopadhyay & Tumulesh K. S. Solanky, 2012. "On Two-Stage Comparisons with a Control Under Heteroscedastic Normal Distributions," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 501-522, September.
    2. Yata, Kazuyoshi & Aoshima, Makoto, 2013. "PCA consistency for the power spiked model in high-dimensional settings," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 334-354.
    3. Jung, Sungkyu & Sen, Arusharka & Marron, J.S., 2012. "Boundary behavior in High Dimension, Low Sample Size asymptotics of PCA," Journal of Multivariate Analysis, Elsevier, vol. 109(C), pages 190-203.
    4. Wang, Shao-Hsuan & Huang, Su-Yun, 2022. "Perturbation theory for cross data matrix-based PCA," Journal of Multivariate Analysis, Elsevier, vol. 190(C).
    5. Yugo Nakayama & Kazuyoshi Yata & Makoto Aoshima, 2020. "Bias-corrected support vector machine with Gaussian kernel in high-dimension, low-sample-size settings," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(5), pages 1257-1286, October.
    6. Borysov, Petro & Hannig, Jan & Marron, J.S., 2014. "Asymptotics of hierarchical clustering for growing dimension," Journal of Multivariate Analysis, Elsevier, vol. 124(C), pages 465-479.
    7. Chung, Hee Cheol & Ahn, Jeongyoun, 2021. "Subspace rotations for high-dimensional outlier detection," Journal of Multivariate Analysis, Elsevier, vol. 183(C).
    8. Shen, Dan & Shen, Haipeng & Marron, J.S., 2013. "Consistency of sparse PCA in High Dimension, Low Sample Size contexts," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 317-333.
    9. Jun Li, 2018. "Asymptotic normality of interpoint distances for high-dimensional data with applications to the two-sample problem," Biometrika, Biometrika Trust, vol. 105(3), pages 529-546.
    10. Miecznikowski Jeffrey C. & Gaile Daniel P. & Chen Xiwei & Tritchler David L., 2016. "Identification of consistent functional genetic modules," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 15(1), pages 1-18, March.
    11. Makoto Aoshima & Kazuyoshi Yata, 2014. "A distance-based, misclassification rate adjusted classifier for multiclass, high-dimensional data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(5), pages 983-1010, October.
    12. Nakayama, Yugo & Yata, Kazuyoshi & Aoshima, Makoto, 2021. "Clustering by principal component analysis with Gaussian kernel in high-dimension, low-sample-size settings," Journal of Multivariate Analysis, Elsevier, vol. 185(C).
    13. Yata, Kazuyoshi & Aoshima, Makoto, 2012. "Effective PCA for high-dimension, low-sample-size data with noise reduction via geometric representations," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 193-215.
    14. Jung, Sungkyu, 2018. "Continuum directions for supervised dimension reduction," Computational Statistics & Data Analysis, Elsevier, vol. 125(C), pages 27-43.
    15. Yata, Kazuyoshi & Aoshima, Makoto, 2010. "Effective PCA for high-dimension, low-sample-size data with singular value decomposition of cross data matrix," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 2060-2077, October.
    16. Modarres, Reza, 2022. "A high dimensional dissimilarity measure," Computational Statistics & Data Analysis, Elsevier, vol. 175(C).
    17. Yata, Kazuyoshi & Aoshima, Makoto, 2013. "Correlation tests for high-dimensional data using extended cross-data-matrix methodology," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 313-331.
    18. Wang, Shao-Hsuan & Huang, Su-Yun & Chen, Ting-Li, 2020. "On asymptotic normality of cross data matrix-based PCA in high dimension low sample size," Journal of Multivariate Analysis, Elsevier, vol. 175(C).
    19. Lee, Myung Hee, 2012. "On the border of extreme and mild spiked models in the HDLSS framework," Journal of Multivariate Analysis, Elsevier, vol. 107(C), pages 162-168.
    20. Bandyopadhyay, Uttam & Sarkar, Suman & Biswas, Atanu, 2022. "Sequential confidence interval for comparing two Bernoulli distributions in a non-conventional set-up," Statistics & Probability Letters, Elsevier, vol. 181(C).

    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:spr:metcap:v:14:y:2012:i:3:d:10.1007_s11009-011-9233-z. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.