Abstract
The asymptotic expansion of the distribution of the gradient test statistic is derived for a composite hypothesis under a sequence of Pitman alternative hypotheses converging to the null hypothesis at rate n −1/2, n being the sample size. Comparisons of the local powers of the gradient, likelihood ratio, Wald and score tests reveal no uniform superiority property. The power performance of all four criteria in one-parameter exponential family is examined.
Similar content being viewed by others
References
Bartlett M.S. (1953a) Approximate confidence intervals. Biometrika 40: 12–19
Bartlett M.S. (1953b) Approximate confidence intervals, II. More than one unknown parameter. Biometrika 40: 306–317
Bracewell R. (1999) The Fourier transform and its applications (3rd ed.). McGraw-Hill, New York
Cordeiro G.M., Botter D.A., Ferrari S.L.P. (1994) Nonnull asymptotic distributions of three classic criteria in generalised linear models. Biometrika 81: 709–720
Cox D.R., Snell E.J. (1968) A general definition of residuals (with discussion). Journal of the Royal Statistical Society B 30: 248–275
Ferrari S.L.P., Botter D.A., Cribari-Neto F. (1997) Local power of three classic criteria in generalised linear models with unknown dispersion. Biometrika 84: 482–485
Firth D. (1993) Bias reduction of maximum likelihood estimates. Biometrika 80: 27–38
Harris P., Peers H.W. (1980) The local power of the efficient score test statistic. Biometrika 67: 525–529
Hayakawa T. (1975) The likelihood ratio criterion for a composite hypothesis under a local alternative. Biometrika 62: 451–460
Hayakawa T. (1977) The likelihood ratio criterion and the asymptotic expansion of its distribution. Annals of the Institute of Statistical Mathematics 29: 359–378
Hayakawa T., Puri M.L. (1985) Asymptotic expansions of the distributions of some test statistics. Annals of the Institute of Statistical Mathematics 37: 95–108
Peers H.W. (1971) Likelihood ratio and associated test criteria. Biometrika 58: 577–587
Rao C.R. (1948) Large sample tests of statistical hypotheses concerning several parameters with applications to problens of estimation. Proceedings of the Cambridge Philosophical Society 44: 50–57
Rao C.R. (2005) Score test: historical review and recent developments. In: Balakrishnan N., Kannan N., Nagaraja H.N. (eds) Advances in ranking and selection, multiple comparisons, and reliability. Birkhauser, Boston
Rao C.R., Mukerjee R. (1997) Comparison of LR, score, and Wald tests in a non-iid setting. Journal of Multivariate Analysis 60: 99–110
Terrell G.R. (2002) The gradient statistic. Computing Science and Statistics 34: 206–215
Wald A. (1943) Tests of statistical hypothesis concerning several parameters when the number of observations is large. Transactions of the American Mathematical Society 54: 426–482
Wilks S.S. (1938) The large-sample distribution of the likelihood ratio for testing composite hypothesis. Annals of Mathematical Statistics 9: 60–62
Author information
Authors and Affiliations
Corresponding author
Additional information
We gratefully acknowledge grants from FAPESP and CNPq (Brazil).
About this article
Cite this article
Lemonte, A.J., Ferrari, S.L.P. The local power of the gradient test. Ann Inst Stat Math 64, 373–381 (2012). https://doi.org/10.1007/s10463-010-0315-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-010-0315-4