The Annals of Statistics

Asymptotically uniformly most powerful tests in parametric and semiparametric models

Sungsub Choi,W. J. Hall, and Anton Schick

Full-text: Open access

Abstract

Tests of hypotheses about finite-dimensional parameters in a semiparametric model are studied from Pitman's moving alternative (or local) approach using Le Cam's local asymptotic normality concept. For the case of a real parameter being tested, asymptotically uniformly most powerful (AUMP) tests are characterized for one-sided hypotheses, and AUMP unbiased tests for two-sided ones. An asymptotic invariance principle is introduced for multidimensional hypotheses, and AUMP invariant tests are characterized. These provide optimality for Wald, Rao (score), Neyman-Rao (effective score) and likelihood ratio tests in parametric models, and for Neyman-Rao tests in semiparametric models when constructions are feasible. Inversions lead to asymptotically uniformly most accurate confidence sets. Examples include one-, two- and k-sample problems, a linear regression model with unknown error distribution and a proportional hazards regression model with arbitrary baseline hazards. Results are presented in a format that facilitates application in strictly parametric models.

Article information

Source
Ann. Statist. Volume 24, Number 2 (1996), 841-861.

Dates
First available: 24 September 2002

Permanent link to this document
http://projecteuclid.org/euclid.aos/1032894469

Mathematical Reviews number (MathSciNet)
MR1394992

Digital Object Identifier
doi:10.1214/aos/1032894469

Zentralblatt MATH identifier
0860.62020

Subjects
Primary: 62F05: Asymptotic properties of tests
Secondary: 62G20: Asymptotic properties

Keywords
Local alternatives effective scores unbiased tests invariance efficient tests adaptation asymptotic confidence sets

Citation

Choi, Sungsub; Hall, W. J.; Schick, Anton. Asymptotically uniformly most powerful tests in parametric and semiparametric models. The Annals of Statistics 24 (1996), no. 2, 841--861. doi:10.1214/aos/1032894469. http://projecteuclid.org/euclid.aos/1032894469.


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References

  • ANDERSEN, P. K., BORGAN, Ø., GILL, R. D. and KEIDING N. 1993. Statistical Models Based on Counting Processes. Springer, New York. Z.
  • BASAWA, I. V. and KOUL, H. L. 1988. Large-sample statistics based on quadratic dispersion. Internat. Statist. Rev. 56 199 219. Z.
  • BEGUN, J. M., HALL, W. J., HUANG, W.-M. and WELLNER, J. A. 1983. Information and asy mptotic efficiency in parametric nonparametric models. Ann. Statist. 11 432 452. Z.
  • BICKEL, P. J. 1982. On adaptive estimation. Ann. Statist. 10 647 671. Z.
  • BICKEL, P. J., KLAASSEN, C. A. J., RITOV, Y. and WELLNER, J. A. 1993. Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins Univ. Press. Z.
  • CHOI, S. 1989. On asy mptotically optimal tests. Ph.D. dissertation, Dept. Statistics, Univ. Rochester. Z.
  • COX, D. R. 1972. Regression models and life tables. J. Roy. Statist. Soc. Ser. B. 34 187 220. Z.
  • COX, D. R. 1975. Partial likelihood. Biometrika 62 269 279. Z.
  • EFRON, B. 1977. The efficiency of Cox's likelihood function for censored data. J. Amer. Statist. Assoc. 72 557 565.Z.
  • FABIAN, V. and HANNAN, J. 1985. Introduction to Probability and Mathematical Statistics. Wiley, New York.Z.
  • FABIAN, V. and HANNAN, J. 1987. Local asy mptotic behavior of densities. Statist. Decisions 5
  • HAJEK, J. 1962. Asy mptotically most powerful rank order tests. Ann. Math. Statist. 33 ´ 1124 1147.
  • HAJEK, J. and SIDAK, J. 1967. Theory of Rank Tests. Academic Press, New York. ´ ´ Z.
  • HALL, W. J. and MATHIASON, D. 1990. On large-sample estimation and testing in parametric models. Internat. Statist. Rev. 58 77 97.
  • HUANG, W.-M. 1982. Parametric estimation when there are nuisance functions. Ph.D. dissertation, Dept. Statistics, Univ. Rochester. Z.
  • KLAASSEN, C. A. S. 1987. Consistent estimation of the influence function of locally asy mptotically linear estimators. Ann. Statist. 15 1548 1562. Z.
  • LE CAM, L. 1960. Locally asy mptotically normal families of distributions. Univ. California Publ. Statist. 3 37 98. Z.
  • LE CAM, L. 1969. Theorie Asy mptotique de la Decision Statistique. Univ. Montreal Press. ´ ´ ´ Z.
  • LE CAM, L. 1986. Asy mptotic Methods in Statistical Decision Theory. Springer, New York. Z.
  • LE CAM, L. and YANG, G. L. 1990. Asy mptotics in Statistics: Some Basic Concepts. Springer, New York. Z.
  • LEHMANN, E. L. 1986. Testing Statistical Hy potheses, 2nd ed. Wiley, New York. Z.
  • MATHIASON, D. 1982. Large sample test procedures in the presence of nuisance parameters. Ph.D. dissertation, Dept. Statistics, Univ. Rochester. Z.
  • NEy MAN, J. 1959. Optimal asy mptotic tests of composite statistical hy potheses. In Probability Z. Z. and Statistics Harald Cramer Volume U. Grenander, ed. 212 234. Wiley, New ´ York. Z.
  • OAKES, D. 1977. The asy mptotic information in censored survival data. Biometrika 64 441 448. Z.
  • ROUSSAS, G. G. 1972. Contiguity of Probability Measures: Some Applications in Statistics. Cambridge Univ. Press. Z.
  • SCHICK, A. 1986. On asy mptotically efficient estimation in semiparametric models. Ann. Statist. 14 1139 1151. Z.
  • SCHICK, A. 1987. A note on the construction of asy mptotically linear estimators. J. Statist.
  • SCHICK, A. 1993. On efficient estimation in regression models. Ann. Statist. 21 1486 1521.
  • SCHICK, A. 1994. Efficient estimation in regression models with unknown scale functions. Math. Methods Statist. 3 171 212. Z.
  • STEIN, C. 1956. Efficient nonparametric testing and estimation. Proc. Third Berkeley Sy mp. Math. Statist. Probab. 1 187 195. Univ. California Press, Berkeley. Z.
  • STRASSER, H. 1985. Mathematical Theory of Statistics. de Gruy ter, Berlin. Z.
  • WALD, A. 1941. Asy mptotically most powerful tests of statistical hy potheses. Ann. Math. Statist. 12 1 19. Z.
  • WALD, A. 1943. Tests of statistical hy potheses concerning several parameters when the number of observations is large. Trans. Amer. Math. Soc. 54 426 482. Z.
  • WEFELMEy ER, W. 1987. Testing hy potheses on independent, not identically distributed models. Z. In Mathematical Statistics and Probability Theory M. L. Puri et al., eds. A 267 282. Reidel, Berlin.
  • MCGILL UNIVERSITY ROCHESTER, NEW YORK 14627 1020 PINE AVENUE WEST
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