Calibration of the empirical likelihood for high-dimensional data
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
This article is concerned with the calibration of the empirical likelihood (EL) for high-dimensional data where the data dimension may increase as the sample size increases. We analyze the asymptotic behavior of the EL under a general multivariate model and provide weak conditions under which the best rate for the asymptotic normality of the empirical likelihood ratio (ELR) is achieved. In addition, there is usually substantial lack-of-fit when the ELR is calibrated by the usual normal in high dimensions, producing tests with type I errors much larger than nominal levels. We find that this is mainly due to the underestimation of the centralized and normalized quantities of the ELR. By examining the connection between the ELR and the classical Hotelling’s \(T\) -square statistic, we propose an effective calibration method which works much better in most situations.
- Bai, Z., Saranadasa, H. (1996). Effect of high dimension: by an example of a two sample problem. Statistics Sinica, 6, 311–329.
- Brown, B. M., Chen, S. X. (1998). Combined and least squares empirical likelihood. Annals of the Institue of Statistical Mathematics, 50, 697–714.
- Chen, J. H., Variyath, A. M., Abraham, B. (2008). Adjusted empirical likelihood and its properties. Journal of Computational and Graphical Statistics, 17, 426–443.
- Chen, S. X., Qin, Y.-L. (2010). A two-sample test for high-dimensional data with applications to gene-set testing. The Annals of Statistics, 38, 808–835.
- Chen, S. X., Peng, L., Qin, Y.-L. (2009). Effects of data dimension on empirical likelihood. Biometrika, 96, 1–12.
- Chen, S. X., Zhang, L.-X., Zhong, P.-S. (2010). Tests for high-dimensional covariance matrices. Journal of American Statistian Assocciation, 105, 810–815.
- DasGupta, A. (2008). Asymptotic theory of statistics and probability. New York: Springer.
- DiCiccio, T. J., Hall, P., Romano, J. P. (1991). Empirical likelihood is Bartlett correctable. The Annals of Statistics, 19, 1053–1061.
- Emerson, S. C., Owen, A. B. (2009). Calibration of the empirical likelihood method for a vector mean. Econometrical Journal of Statisitcs, 3, 1161–1192.
- Hall, P. (1992). The bootstrap and Edgeworth expansion. New York: Springer.
- Hall, P., Hyde, C. C. (1992). Martingale central limit theory and its applications. New York: Academic Press.
- Hjort, H. L., Mckeague, I. W., Van Keilegom, I. (2009). Extending the scope of empirical likelihood. The Annals of Statistics, 37, 1079–1115.
- Liu, Y., Chen, J. (2010). Adjusted Empirical Likelihood with High-Order Precision. The Annals of Statistics, 38, 1341–1362.
- Liu, Y., Yu, C. W. (2010). Bartlett Correctable Two-Sample Adjusted Empirical Likelihood. Journal of Multivariate Analysis, 101, 1701–1711.
- Mardia, K. V., Kent, J. T., Bibby, J. M. (1979). Multivariate analysis. London: Academic Press.
- Owen, A. B. (1988). Empirical likelihood ratio confidence intervals for a single functional. Biometrika, 75, 237–249.
- Owen, A. B. (1990). Empirical likelihood ratio confidence regions. The Annals of Statistics, 18, 90–120.
- Owen, A. B. (2001). Empirical likelihood. New York: Chapman & Hall/CRC.
- Portnoy, S. (1985). Asymptotic behavior of M-estimations of \(p\) regression parameters with \(p^2/n\) is large. II. normal approximation. The Annals of Statistics, 13, 1403–1417.
- Qin, J., Lawless, J. (1994). Empirical likelihood and general equations. The Annals of Statistics, 22, 300–325.
- Schott, J. R. (2005). Testing for complete independence in high dimensions. Biometrika, 92, 951–956.
- Tang, C. Y., Leng, C. (2005). Penalized high-dimensional empirical likelihood. Biometrika, 97, 905–920.
- Tsao, M. (2004). Bounds on coverage probabilities of the empirical likelihood ratio confidence regions. The Annals of Statistics, 32, 1215–1221.
- Vapnik, V., Chervonenkis, A. (1971). On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and its Applications, 16, 264–280.
- Calibration of the empirical likelihood for high-dimensional data
Annals of the Institute of Statistical Mathematics
Volume 65, Issue 3 , pp 529-550
- Cover Date
- Print ISSN
- Online ISSN
- Springer Japan
- Additional Links
- Asymptotic normality
- Coverage accuracy
- High-dimensional data
- Hotelling’s $$T$$ -square statistic
- Industry Sectors