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Given a random sample of size n from a p dimensional random vector, we are interested in testing whether the p components of the random vector are mutually independent. This is the so-called complete independence test. In the multivariate normal case, it is equivalent to testing whether the correlation matrix is an identity matrix. In this paper, we propose a one-sided empirical likelihood method for the complete independence test based on squared sample correlation coefficients. The limiting distribution for our one-sided empirical likelihood test statistic is proved to be Z^2I(Z > 0) when both n and p tend to infinity, where Z is a standard normal random variable. In order to improve the power of the empirical likelihood test statistic, we also introduce a rescaled empirical likelihood test statistic. We carry out an extensive simulation study to compare the performance of the rescaled empirical likelihood method and two other statistics.
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Empirical likelihood test for a large-dimensional mean vector
Summary This paper is concerned with empirical likelihood inference on the population mean when the dimension $$p$$ and the sample size $$n$$ satisfy $$p/n\rightarrow c\in [1,\infty)$$. As shown in Tsao (2004), the empirical likelihood method fails with high probability when $p/n>1/2$ because the convex hull of the $$n$$ observations in $$\mathbb{R}^p$$ becomes too small to cover the true mean value. Moreover, when $p> n$, the sample covariance matrix becomes singular, and this results in the breakdown of the first sandwich approximation for the log empirical likelihood ratio. To deal with these two challenges, we propose a new strategy of adding two artificial data points to the observed data. We establish the asymptotic normality of the proposed empirical likelihood ratio test. The proposed test statistic does not involve the inverse of the sample covariance matrix. Furthermore, its form is explicit, so the test can easily be carried out with low computational cost. Our numerical comparison shows that the proposed test outperforms some existing tests for high-dimensional mean vectors in terms of power. We also illustrate the proposed procedure with an empirical analysis of stock data.
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- PAR ID:
- 10217333
- Date Published:
- Journal Name:
- Biometrika
- Volume:
- 107
- Issue:
- 3
- ISSN:
- 0006-3444
- Page Range / eLocation ID:
- 591 to 607
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/biomet/asaa005
