Calibration of the empirical likelihood method for a vector mean. (English) Zbl 1326.62099

Summary: The empirical likelihood method is a versatile approach for testing hypotheses and constructing confidence regions in a non-parametric setting. For testing the value of a vector mean, the empirical likelihood method offers the benefit of making no distributional assumptions beyond some mild moment conditions. However, in small samples or high dimensions the method is very poorly calibrated, producing tests that generally have a much higher type I error than the nominal level, and it suffers from a limiting convex hull constraint. Methods to address the performance of the empirical likelihood in the vector mean setting have been proposed in a number of papers, including a contribution that suggests supplementing the observed dataset with an artificial data point. We examine the consequences of this approach and describe a limitation of their method that we have discovered in settings when the sample size is relatively small compared with the dimension. We propose a new modification to the extra data approach that involves adding two points and changing the location of the extra points. We explore the benefits that this modification offers, and show that it results in better calibration, particularly in difficult cases. This new approach also results in a small-sample connection between the modified empirical likelihood method and Hotelling’s \(T\)-square test. We show that varying the location of the added data points creates a continuum of tests that range from the unmodified empirical likelihood statistic to Hotelling’s \(T\)-square statistic.


62G10 Nonparametric hypothesis testing
62H15 Hypothesis testing in multivariate analysis
Full Text: DOI Euclid


[1] Bartolucci, F. (2007). A penalized version of the empirical likelihood ratio for the population mean., Statistics & Probability Letters 77 104-110. · Zbl 1106.62050
[2] Chen, J., Variyath, A. M. and Abraham, B. (2008). Adjusted empirical likelihood and its properties., Journal of Computational and Graphical Statistics 17 426:443. · doi:10.1198/106186008X321068
[3] DiCiccio, T., Hall, P. and Romano, J. (1991). Empirical likelihood is Bartlett-correctable., The Annals of Statistics 19 1053-1061. · Zbl 0725.62042 · doi:10.1214/aos/1176348137
[4] Hotelling, H. (1931). The generalization of Student’s ratio., The Annals of Mathematical Statistics 2 360-378. · Zbl 0004.26503 · doi:10.1214/aoms/1177732979
[5] Kiefer, J. and Schwartz, R. (1965). Admissible Bayes character of, T 2 - , R 2 - , and other fully invariant tests for classical multivariate normal problems. The Annals of Mathematical Statistics 34 747-770. · Zbl 0137.36605 · doi:10.1214/aoms/1177700051
[6] Liu, Y. and Chen, J. (2009). Adjusted empirical likelihood with high-order precision., Annals of Statistics (to appear). · Zbl 1189.62054 · doi:10.1214/09-AOS750
[7] Owen, A. (1988). Empirical likelihood ratio confidence intervals for a single functional., Biometrika 75 237-249. · Zbl 0641.62032 · doi:10.1093/biomet/75.2.237
[8] Owen, A. (1990). Empirical likelihood ratio confidence regions., The Annals of Statistics 18 90-120. · Zbl 0712.62040 · doi:10.1214/aos/1176347494
[9] Owen, A. (2001)., Empirical likelihood . Chapman & Hall/CRC, New York. · Zbl 0989.62019
[10] Stein, C. (1956). The admissibility of Hotelling’s, T 2 -test. The Annals of Mathematical Statistics 27 616-623. · Zbl 0073.14301 · doi:10.1214/aoms/1177728171
[11] Tsao, M. (2001). A small sample calibration method for the empirical likelihood ratio., Statistics & Probability Letters 54 41-45. · Zbl 1052.62054
[12] Tsao, M. (2004a). Bounds on coverage probabilities of the empirical likelihood ratio confidence regions., The Annals of Statistics 32 1215-1221. · Zbl 1091.62040 · doi:10.1214/009053604000000337
[13] Tsao, M. (2004b). A new method of calibration for the empirical loglikelihood ratio., Statistics & Probability Letters 68 305-314. · Zbl 1075.62012
[14] Wendel, J. G. (1962). A problem in geometric probability., Mathematica Scandinavica 11 109-111. · Zbl 0108.31603
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