Electronic Journal of Statistics

Gradient statistic: Higher-order asymptotics and Bartlett-type correction

Tiago M. Vargas, Silvia L.P. Ferrari, and Artur J. Lemonte

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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 [10]. 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.

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Electron. J. Statist., Volume 7 (2013), 43-61.

First available in Project Euclid: 11 January 2013

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Asymptotic expansion Bartlett-type correction Bayesian route gradient test shrinkage argument


Vargas, Tiago M.; Ferrari, Silvia L.P.; Lemonte, Artur J. Gradient statistic: Higher-order asymptotics and Bartlett-type correction. Electron. J. Statist. 7 (2013), 43--61. doi:10.1214/12-EJS763. https://projecteuclid.org/euclid.ejs/1357913281

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  • [1] Bai, P. (2009). Sphericity test in a GMANOVA-MANOVA model with normal error., Journal of Multivariate Analysis 100, 2305–2312.
  • [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.
  • [3] Birnbaum, Z.W., Saunders, S.C. (1969). A new family of life distributions., Journal of Applied Probability 6, 319–327.
  • [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.
  • [5] Chang, H.I., Mukerjee, R. (2011). Data-dependent probability matching priors for likelihood ratio and adjusted likelihood ratio statistics., Statistics. In press, DOI:10.1080/02331888.2011.587880.
  • [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.
  • [7] Cordeiro, G.M., Cribari-Neto, F. (1996). On Bartlett and Bartlett-type corrections., Econometric Reviews 15, 339–367.
  • [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.
  • [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.
  • [11] Harris, P. (1985). An asymptotic expansion for the null distribution of the efficient score statistic., Biometrika 72, 653–659.
  • [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.
  • [13] Hill, G.W., Davis, A.W. (1968). Generalized asymptotic expansions of Cornish–Fisher type., The Annals of Mathematical Statistics 39, 1264-73.
  • [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.
  • [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.
  • [16] Lawley, D. (1956). A general method for approximating to the distribution of likelihood ratio criteria., Biometrika 43, 295–303.
  • [17] Lemonte, A.J. (2011). Local power of some tests in exponential family nonlinear models., Journal of Statistical Planning and Inference 141, 1981–1989.
  • [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.
  • [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.
  • [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.
  • [21] Mukerjee, R., Reid, N. (2000). On the Bayesian approach for frequentist computations., Brazilian Journal of Probability and Statistics 14, 159–166.
  • [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.
  • [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.
  • [27] van Giersbergen, N.P.A. (2009). Bartlett correction in the stable AR(1) model with intercept and trend., Econometric Theory 25, 857–872.
  • [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.
  • [29] Wilks, S.S. (1938). The large-sample distribution of the likelihood ratio for testing composite hypothesis., Annals of Mathematical Statistics 9, 60–62.
  • [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.