Abstract
We derive the exact Bahadur slopes of studentized score tests for a simple null hypothesis in a one-parameter family of distributions. The Student's t-test is included as a special case for which a recent result of Rukhin (1993, Sankhyā Ser. A, 55, 159–163) was improved upon. It is shown that locally optimal Bahadur efficiency for one-sample location models with a known or estimated scale parameter is attained within the class of studentized score tests. The studentized test has an asymptotic null distribution free of the scale parameter, and the optimality of likelihood scores does not depend on the existence of a moment generating function. We also consider the influence function and breakdown point of such tests as part of our robustness investigation. The influence of any studentized score test is bounded from above, indicating certain degree of robustness of validity, but a bounded score function is needed to cap the influence from below and to ensure a high power breakdown point. We find that the standard Huber-type score tests are not only locally minimax in Bahadur efficiency, but also very competitive in global efficiency at a variety of location models.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Akritas, M. G. and Kourouklis, S. (1988). Local Bahadur efficiency of score tests, J. Statist. Plann. Inference, 19, 187–199.
Bahadur, R. R. (1960). Stochastic comparison of tests, Ann. Statist. 31, 276–295.
Bahadur, R. R. (1965). An optimal property of the likelihood ratio statistic, Proc. Fifth Berkeley Symp. Math. Statist. Prob., 1, 13–26.
Bahadur, R. R. (1971). Some Limit Theorems in Statistics, SIAM, Philadelphia.
Efron, B. (1969). Student's t-test under symmetry conditions, J. Amer. Statist. Assoc., 64, 1278–1302.
Griffin, P. S. and Kuelbs, J. (1989). Self-normalized laws of the iterated logarithm, Ann. Probab., 17, 1571–1601.
Groeneboom, P. and Oosterhoff, J. (1977). Bahadur efficiency and probability of large deviations, Statist. Neerlandica, 31, 1–24.
Hampel, F. R. (1974). The influence and its role in robust estimation. J. Amer. Statist. Assoc., 69, 383–393.
Hampel, F. R., Ronchetti, E., Rousseeuw, P. J. and Stahel, W. A. (1986). Robust Statistics: The Approach Based on Influence Functions, Wiley, New York.
He, X, Simpson, D. G. and Portnoy, S. L. (1990). Breakdown robustness of tests, J. Amer. Statist. Assoc., 85, 446–452.
Huber, P. J. (1964). Robust estimation of a location parameter, Ann. Math. Statist., 35, 73–101.
Huber, P. J. (1965). A robust version of the probability ratio test, Ann. Math. Statist., 36, 1753–1758.
Kallenberg, W. C. M. and Ledwina, T. (1987). On local and nonlocal measures of efficiency, Ann. Statist. 15, 1401–1420.
Lambert, D. (1981). Influence functions for tests, J. Amer. Statist. Assoc., 76, 649–657.
Lambert, D. and Hall, W. J. (1982). Asymptotic lognormality of P-values, Ann. Statist., 10, 44–64.
Lloyd, C. (1994). Approximate pivots from M-estimators, Statistica Sinica, 4, 701–714.
Rukhin, A. L. (1993). On the Bahadur efficiency of a t-test, Sankhyā Ser. A, 55, 159–163.
Shao, Q. M. (1994). Self-normalized Large Deviations, Ann. Probab. (to appear in 1997).
Author information
Authors and Affiliations
About this article
Cite this article
He, X., Shao, Qm. Bahadur efficiency and robustness of studentized score tests. Ann Inst Stat Math 48, 295–314 (1996). https://doi.org/10.1007/BF00054792
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00054792