Open Access
April 2010 Asymptotic inference for high-dimensional data
Jim Kuelbs, Anand N. Vidyashankar
Ann. Statist. 38(2): 836-869 (April 2010). DOI: 10.1214/09-AOS718


In this paper, we study inference for high-dimensional data characterized by small sample sizes relative to the dimension of the data. In particular, we provide an infinite-dimensional framework to study statistical models that involve situations in which (i) the number of parameters increase with the sample size (that is, allowed to be random) and (ii) there is a possibility of missing data. Under a variety of tail conditions on the components of the data, we provide precise conditions for the joint consistency of the estimators of the mean. In the process, we clarify and improve some of the recent consistency results that appeared in the literature. An important aspect of the work presented is the development of asymptotic normality results for these models. As a consequence, we construct different test statistics for one-sample and two-sample problems concerning the mean vector and obtain their asymptotic distributions as a corollary of the infinite-dimensional results. Finally, we use these theoretical results to develop an asymptotically justifiable methodology for data analyses. Simulation results presented here describe situations where the methodology can be successfully applied. They also evaluate its robustness under a variety of conditions, some of which are substantially different from the technical conditions. Comparisons to other methods used in the literature are provided. Analyses of real-life data is also included.


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Jim Kuelbs. Anand N. Vidyashankar. "Asymptotic inference for high-dimensional data." Ann. Statist. 38 (2) 836 - 869, April 2010.


Published: April 2010
First available in Project Euclid: 19 February 2010

zbMATH: 1184.62094
MathSciNet: MR2604698
Digital Object Identifier: 10.1214/09-AOS718

Primary: 60B10 , 60B12 , 60F05 , 62A01 , 62F40 , 62G20 , 62H15 , 92B15

Keywords: c_0 , covariance matrix estimation , functional genomics , High-dimensional data , infinite-dimensional central limit theorem , joint inference , l_ρ , large p small n , laws of large numbers , microarrays , shrinkage , structured covariance matrices

Rights: Copyright © 2010 Institute of Mathematical Statistics

Vol.38 • No. 2 • April 2010
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