Abstract
This paper is concerned with the theory of testing hypothesis with composite null hypothesis or with nuisance parameters. The asymptotic behaviour of the likelihood ratio and the associated test statistics are investigated. Under a class of local alternatives with local orthogonality relative to the nuisance parameter vector, a unique decomposition of local power is presented. The decomposition consists of two parts; one is the influence of nuisance parameters and the other is the power corresponding to the simple case where the nuisance parameters are known. The decomposition formula is applied to some examples, including the gamma, Weibull and location-scale family.
Similar content being viewed by others
References
Amari, S. (1985). Differential-Geometrical Methods in Statistics, Springer, New York.
Cox, D. R. and Hinkley, D. V. (1974). Theoretical Statistics, Chapman and Hall, London.
Cox, D. R. and Reid, N. (1987). Parameter orthogonality and approximate conditional inference, J. Roy. Statist. Soc. Ser. B, 49, 1–39.
Eguchi, S. (1983). Second order eficiency of minimum contrast estimator in a curved exponential family, Ann. Statist., 11, 793–803.
Eguchi, S. (1985). A differential geometric approach to statistical inference on the basis of contrast functionals, Hiroshima Math., 15, 341–391.
Eguchi, S. (1987). A geometric look at local power comparison of test statistics, Statistical Research Group of Hiroshima Univ., TR 202.
Finney, D. J. (1976). Radioligand assay, Biometrics, 32, 721–740.
Harris, P. and Peers, H. W. (1980). The local power of eficient scores test statistics, Biometrika, 67, 525–529.
Hayakawa, T. (1975). The likelihood ratio criterion for a composite hypothesis under a local alternative, Biometrika, 58, 577–587.
Kumon, M. and Amari, S. (1983). Geometrical theory of higher-order asymptotics of test, interval estimator, conditional inference, Proc. Roy. Soc. London Ser. A, 387, 429–458.
Kumon, M. and Amari, S. (1988). Differntial geometry of testing hypothesis—A higher order asymptotic theory in multi-parameter curved exponential family, J. Fac. Engrg. Univ. Tokyo, 36.
McCullagh, P. and Cox, D. R. (1986). Invariants and likelihood ratio statistics, Ann. Statist., 14, 1419–1430.
Nagao, H. (1974). Asymptotic nonnull distributions of certain test criteria for a covariance matrix, J. Multivariate Anal., 4, 409–418.
Peers, H. W. (1971). Likelihood ratio and associated test criteria, Biometrika, 58, 577–587.
Sugiura, N. (1973). Asymptotic non-null distribution of the likelihood ratio criteria for covariance matrix under local alternatives, Ann. Statist., 1, 718–728.
Voss, P. W. (1989). Fundamental equations for statistical submanifolds with applications to the bartlett corrections. Ann. Inst. Statist. Math., 41, 429–450.
Author information
Authors and Affiliations
About this article
Cite this article
Eguchi, S. A geometric look at nuisance parameter effect of local powers in testing hypothesis. Ann Inst Stat Math 43, 245–260 (1991). https://doi.org/10.1007/BF00118634
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00118634