×

zbMATH — the first resource for mathematics

Gradient statistic: higher-order asymptotics and Bartlett-type correction. (English) Zbl 1336.62143
Summary: We obtain an asymptotic expansion for the null distribution function of the gradient statistic for testing composite null hypotheses in the presence of nuisance parameters. The expansion is derived using a Bayesian route based on the shrinkage argument described in [J. K. Ghosh and R. Mukerjee, J. Multivariate Anal. 38, No. 2, 385–393 (1991; Zbl 0728.62020)]. Using this expansion, we propose a Bartlett-type corrected gradient statistic with chi-square distribution up to an error of order \(o(n^{-1})\) under the null hypothesis. Further, we also use the expansion to modify the percentage points of the large sample reference chi-square distribution. Monte Carlo simulation experiments and various examples are presented and discussed.

MSC:
62H15 Hypothesis testing in multivariate analysis
62J07 Ridge regression; shrinkage estimators (Lasso)
PDF BibTeX XML Cite
Full Text: DOI Euclid arXiv
References:
[1] Bai, P. (2009). Sphericity test in a GMANOVA-MANOVA model with normal error., Journal of Multivariate Analysis 100 , 2305-2312. · Zbl 1176.62059
[2] Bickel, P.J., Ghosh, J.K. (1990). A decomposition for the likelihood ratio statistic and the Bartlett correction - a Bayesian argument., Annals of Statistics 18 , 1070-1090. · Zbl 0727.62035
[3] Birnbaum, Z.W., Saunders, S.C. (1969). A new family of life distributions., Journal of Applied Probability 6 , 319-327. · Zbl 0209.49801
[4] Chang, H.I., Mukerjee, R. (2010). Highest posterior density regions with approximate frequentist validity: the role of data-dependent priors., Statistics and Probability Letters 80 , 1791-1797. · Zbl 1202.62039
[5] Chang, H.I., Mukerjee, R. (2011). Data-dependent probability matching priors for likelihood ratio and adjusted likelihood ratio statistics., Statistics . · Zbl 1440.62094
[6] Cordeiro, G.M., Ferrari, S.L.P. (1991). A modified score test statistic having chi-squared distribuition to order \(n^-1\)., Biometrika 78 , 573-582. · Zbl 1192.62053
[7] Cordeiro, G.M., Cribari-Neto, F. (1996). On Bartlett and Bartlett-type corrections., Econometric Reviews 15 , 339-367. · Zbl 0885.62021
[8] Cox, D.R., Reid, N. (1987). Parameter orthogonality and approximate conditional inference (with discussion)., Journal of the Royal Statistical Society B 40 , 1-39. · Zbl 0616.62006
[9] Datta, G.S., Mukerjee, R. (2003)., Probability Matching Priors: Higher Order Asymptoptics . Springer-Verlag: New York.
[10] Ghosh, J.K., Mukerjee, R. (1991). Characterization of priors under wich Bayesian and frequentist Bartlett corrections are equivalent in the multiparameter case., Journal of Multivariate Analysis 38 , 385-393. · Zbl 0728.62020
[11] Harris, P. (1985). An asymptotic expansion for the null distribution of the efficient score statistic., Biometrika 72 , 653-659. · Zbl 0586.62034
[12] Hayakawa, T. (1977). The likelihood ratio criterion and the asymptotic expansion of its distribution., Annals of the Institute of Statistical Mathematics 29 , 359-378. · Zbl 0438.62015
[13] Hill, G.W., Davis, A.W. (1968). Generalized asymptotic expansions of Cornish-Fisher type., The Annals of Mathematical Statistics 39 , 1264-73. · Zbl 0162.22404
[14] Lagos, B.M., Morettin, P.A. (2004). Improvement of the likelihood ratio test statistic in ARMA models., Journal of Time Series Analysis 25 , 83-101. · Zbl 1051.62079
[15] Lagos, B.M., Morettin, P.A., Barroso, L.P. (2010). Some corrections of the score test statistic for gaussian ARMA models., Brazilian Journal of Probability and Statistics 24 , 434-456. · Zbl 1298.62154
[16] Lawley, D. (1956). A general method for approximating to the distribution of likelihood ratio criteria., Biometrika 43 , 295-303. · Zbl 0073.13602
[17] Lemonte, A.J. (2011). Local power of some tests in exponential family nonlinear models., Journal of Statistical Planning and Inference 141 , 1981-1989. · Zbl 1394.62089
[18] Lemonte, A.J. (2012). Local power properties of some asymptotic tests in symmetric linear regression models., Journal of Statistical Planning and Inference 142 , 1178-1188. · Zbl 1236.62007
[19] Lemonte, A.J., Ferrari, S.L.P. (2012a). The local power of the gradient test., Annals of the Institute of Statistical Mathematics 64 , 373-381. · Zbl 1238.62018
[20] Lemonte, A.J., Ferrari, S.L.P. (2012b). A note on the local power of the LR, Wald, score and gradient tests., Electronic Journal of Statistics 6 , 421-434. · Zbl 1336.62061
[21] Mukerjee, R., Reid, N. (2000). On the Bayesian approach for frequentist computations., Brazilian Journal of Probability and Statistics 14 , 159-166. · Zbl 0983.62014
[22] Noma, H. (2011). Confidence intervals for a random-effects meta-analysis based on Bartlett-type corrections., Statistics in Medicine 30 , 3304-3312.
[23] 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. · Zbl 0034.07503
[24] Rao, C.R. (2005). Score test: historical review and recent developments. In, Advances in Ranking and Selection, Multiple Comparisons, and Reliability , N. Balakrishnan, N. Kannan and H. N. Nagaraja, eds. Birkhuser, Boston.
[25] Terrell, G.R. (2002). The gradient statistic., Computing Science and Statistics 34 , 206-215.
[26] Tu, D., Chen, J., Shi, P., Wu, Y. (2005). A Bartlett type correction for Rao’s score test in Cox regression model., Sankhya 67 , 722-735. · Zbl 1192.62055
[27] van Giersbergen, N.P.A. (2009). Bartlett correction in the stable AR(1) model with intercept and trend., Econometric Theory 25 , 857-872. · Zbl 1253.62066
[28] 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. · Zbl 0063.08120
[29] Wilks, S.S. (1938). The large-sample distribution of the likelihood ratio for testing composite hypothesis., Annals of Mathematical Statistics 9 , 60-62. · Zbl 0018.32003
[30] Zucker, D.M., Lieberman, O., Manor, O. (2000). Improved small sample inference in the mixed linear model: Bartlett correction and adjusted likelihood., Journal of the Royal Statistical Society B 62 , 827-838. · Zbl 0966.62041
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.