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Saddlepoint approximations for likelihood ratio like statistics with applications to permutation tests. (English) Zbl 1246.62121

Summary: We obtain two theorems extending the use of a saddlepoint approximation to multiparameter problems for likelihood ratio-like statistics which allow their use in permutation and rank tests and could be used in bootstrap approximations. In the first, we show that in some cases when no density exists, the integral of the formal saddlepoint density over the set corresponding to large values of the likelihood ratio-like statistic approximates the true probability with relative error of order \(1/n\). In the second, we give multivariate generalizations of the Lugannani-Rice and Barndorff-Nielsen or \(r*\) formulas for the approximations. These theorems are applied to obtain permutation tests based on the likelihood ratio-like statistics for the \(k\) sample and the multivariate two-sample cases. Numerical examples are given to illustrate the high degree of accuracy, and these statistics are compared to the classical statistics in both cases.

MSC:

62G10 Nonparametric hypothesis testing
62H12 Estimation in multivariate analysis
62G20 Asymptotic properties of nonparametric inference
60F10 Large deviations

Software:

PATSYM; bootlib
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References:

[1] Albers, W., Bickel, P. J. and van Zwet, W. R. (1976). Asymptotic expansions for the power of distribution free tests in the one-sample problem. Ann. Statist. 4 108-156. · Zbl 0321.62049
[2] Barndorff-Nielsen, O. E. (1986). Inference on full or partial parameters based on the standardized signed log likelihood ratio. Biometrika 73 307-322. · Zbl 0605.62020
[3] Barndorff-Nielsen, O. E. and Cox, D. R. (1984). Bartlett adjustments to the likelihood ratio statistic and the distribution of the maximum likelihood estimator. J. Roy. Statist. Soc. Ser. B 46 483-495. · Zbl 0581.62016
[4] Borovkov, A. A. and Rogozin, B. A. (1965). On the multi-dimensional central limit theorem. Theory Probab. Appl. 10 55-62. · Zbl 0139.35206
[5] Cramér, H. (1938). Sur un noveau théorème-limite de la théorie des probabilitiés. Actualités Schi. Indust 736 5-23. · JFM 64.0529.01
[6] Davison, A. C. and Hinkley, D. V. (1997). Bootstrap Methods and Their Application. Cambridge Series in Statistical and Probabilistic Mathematics 1 . Cambridge Univ. Press, Cambridge. · Zbl 0886.62001
[7] Genz, A. (2003). Fully symmetric interpolatory rules for multiple integrals over hyper-spherical surfaces. J. Comput. Appl. Math. 157 187-195. · Zbl 1028.65015
[8] Jing, B. Y., Feuerverger, A. and Robinson, J. (1994). On the bootstrap saddlepoint approximations. Biometrika 81 211-215. · Zbl 0796.62015
[9] Osipov, L. V. (1981). On large deviations for sums of random vectors in R k . J. Multivariate Anal. 11 115-126. · Zbl 0457.60019
[10] Robinson, J. (2004). Multivariate tests based on empirical saddlepoint approximations. Metron 62 1-14.
[11] Robinson, J., Höglund, T., Holst, L. and Quine, M. P. (1990). On approximating probabilities for small and large deviations in R d . Ann. Probab. 18 727-753. · Zbl 0704.60018
[12] Robinson, J., Ronchetti, E. and Young, G. A. (2003). Saddlepoint approximations and tests based on multivariate M -estimates. Ann. Statist. 31 1154-1169. · Zbl 1056.62023
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