Statistical Science

Test for a Mean Vector with Fixed or Divergent Dimension

Liang Peng, Yongcheng Qi, and Fang Wang

Full-text: Open access

Abstract

It has been a long history in testing whether a mean vector with a fixed dimension has a specified value. Some well-known tests include the Hotelling $T^{2}$-test and the empirical likelihood ratio test proposed by Owen [Biometrika 75 (1988) 237–249; Ann. Statist. 18 (1990) 90–120]. Recently, Hotelling $T^{2}$-test has been modified to work for a high-dimensional mean, and the empirical likelihood method for a mean has been shown to be valid when the dimension of the mean vector goes to infinity. However, the asymptotic distributions of these tests depend on whether the dimension of the mean vector is fixed or goes to infinity. In this paper, we propose to split the sample into two parts and then to apply the empirical likelihood method to two equations instead of d equations, where d is the dimension of the underlying random vector. The asymptotic distribution of the new test is independent of the dimension of the mean vector. A simulation study shows that the new test has a very stable size with respect to the dimension of the mean vector, and is much more powerful than the modified Hotelling $T^{2}$-test.

Article information

Source
Statist. Sci., Volume 29, Number 1 (2014), 113-127.

Dates
First available in Project Euclid: 9 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.ss/1399645740

Digital Object Identifier
doi:10.1214/13-STS425

Mathematical Reviews number (MathSciNet)
MR3201858

Zentralblatt MATH identifier
1332.62180

Keywords
Empirical likelihood high-dimensional mean test

Citation

Peng, Liang; Qi, Yongcheng; Wang, Fang. Test for a Mean Vector with Fixed or Divergent Dimension. Statist. Sci. 29 (2014), no. 1, 113--127. doi:10.1214/13-STS425. https://projecteuclid.org/euclid.ss/1399645740


Export citation

References

  • Arlot, S., Blanchard, G. and Roquain, E. (2010a). Some nonasymptotic results on resampling in high dimension. I. Confidence regions. Ann. Statist. 38 51–82.
  • Arlot, S., Blanchard, G. and Roquain, E. (2010b). Some nonasymptotic results on resampling in high dimension. II. Multiple tests. Ann. Statist. 38 83–99.
  • Bai, Z. and Saranadasa, H. (1996). Effect of high dimension: By an example of a two sample problem. Statist. Sinica 6 311–329.
  • Chen, S. X., Peng, L. and Qin, Y. (2009). Empirical likelihood methods for high dimension. Biometrika 96 711–722.
  • Chen, S. X. and Qin, Y.-L. (2010). A two-sample test for high-dimensional data with applications to gene-set testing. Ann. Statist. 38 808–835.
  • Chen, S. X. and Van Keilegom, I. (2009). A review on empirical likelihood methods for regression. TEST 18 415–447.
  • Dharmadhikari, S. W. and Jogdeo, K. (1969). Bounds on moments of certain random variables. Ann. Math. Statist. 40 1506–1509.
  • Hall, P. and La Scala, B. (1990). Methodology and algorithms of empirical likelihood. Internat. Statist. Rev. 58 109–127.
  • Hjort, N. L., McKeague, I. W. and Van Keilegom, I. (2009). Extending the scope of empirical likelihood. Ann. Statist. 37 1079–1111.
  • Kuelbs, J. and Vidyashankar, A. N. (2010). Asymptotic inference for high-dimensional data. Ann. Statist. 38 836–869.
  • Owen, A. B. (1988). Empirical likelihood ratio confidence intervals for a single functional. Biometrika 75 237–249.
  • Owen, A. (1990). Empirical likelihood ratio confidence regions. Ann. Statist. 18 90–120.
  • Owen, A. (2001). Empirical Likelihood. Chapman & Hall/CRC, Boca Raton, FL.
  • Srivastava, M. S. (2009). A test for the mean vector with fewer observations than the dimension under non-normality. J. Multivariate Anal. 100 518–532.
  • Srivastava, M. S. and Du, M. (2008). A test for the mean vector with fewer observations than the dimension. J. Multivariate Anal. 99 386–402.
  • von Bahr, B. and Esseen, C. G. (1965). Inequality for the $r$th absolute moment of a sum of random variables, $1\le r\le2$. Ann. Math. Statist. 36 299–393.
  • Yin, Y. Q., Bai, Z. D. and Krishnaiah, P. R. (1988). On the limit of the largest eigenvalue of the large-dimensional sample covariance matrix. Probab. Theory Related Fields 78 509–521.