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On the efficiency of affine invariant multivariate rank tests. (English) Zbl 1127.62361

Summary: In this paper the asymptotic Pitman efficiencies of the affine invariant multivariate analogues of the rank tests based on the generalized median of Oja are considered. Formulae for asymptotic relative efficiencies are found and, under multivariate normal and multivariate \(t\)-distributions, relative efficiencies with respect to Hotelling’s \(T^2\)-test are calculated.

MSC:

62H15 Hypothesis testing in multivariate analysis
62G10 Nonparametric hypothesis testing
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[1] Brown, B. M.; Hettmansperger, T. P., Affine invariant rank methods in the bivariate location model, J. Roy. Statist. Soc. Ser. B, 49, 301-310 (1997) · Zbl 0653.62039
[2] Brown, B. M.; Hettmansperger, T. P., Invariant tests in bivariate models and the \(L_1\), Statistical Data Analysis Based on the \(L_1 (1987)\), North-Holland: North-Holland Amsterdam, p. 333-344 · Zbl 0653.62039
[3] Brown, B. M.; Hettmansperger, T. P., An affine invariant bivariate version of the sign test, J. Roy. Statist. Soc. Ser. B, 51, 117-125 (1989) · Zbl 0675.62036
[4] Chakraborty, B.; Chaudhuri, P., On a transformation and retransformation technique for constructing affine equivariant multivariate median, Proc. Amer. Math. Soc., 124, 2539-2547 (1996) · Zbl 0856.62046
[5] Chakraborty, B.; Chaudhuri, P., On an adaptative transformation and retransformation estimate of multivariate location, J. Roy. Statist. Soc. Ser. B, 60, 145-157 (1998) · Zbl 0909.62056
[6] Chakraborty, B.; Chaudhuri, P.; Oja, H., Operating transformation retransformation on spatial median and angle test, Statist. Sinica (1998) · Zbl 0915.62051
[7] Chaudhuri, P., Multivariate location estimation using extension of \(RU\), Ann. Statist., 20, 897-916 (1992) · Zbl 0762.62013
[8] Dahel, S.; Giri, N., Some distributions related to a noncentral Wishart distribution, Comm. Statist. Theory Methods, 23, 229-237 (1994) · Zbl 0825.62080
[9] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G., Higher Transcendental Functions (1953), McGraw-Hill: McGraw-Hill New York · Zbl 0051.30303
[10] Hajek, J.; Sidak, Z., Theory of Rank tests (1967), Academic Press: Academic Press San Diego · Zbl 0161.38102
[11] Hettmansperger, T. P.; Möttönen, J.; Oja, H., Affine invariant multivariate one-sample signed-rank tests, J. Amer. Statist. Assoc., 92, 1591-1600 (1997) · Zbl 0943.62051
[12] Hettmansperger, T. P.; Möttönen, J.; Oja, H., Affine invariant multivariate rank tests for several samples, Statistica Sinica (1998) · Zbl 0905.62062
[13] Hettmansperger, T. P.; Nyblom, J.; Oja, H., Affine invariant multivariate one-sample sign tests, J. Roy. Statist. Soc. Ser. B, 56, 221-234 (1994) · Zbl 0795.62055
[14] Hettmansperger, T. P.; Oja, H., Affine invariant multivariate multisample sign tests, J. Roy. Statist. Soc. Ser. B, 56, 235-249 (1994) · Zbl 0795.62056
[15] Hössjer, O.; Croux, C., Generalizing univariate signed rank statistics for testing and estimating a multivariate location parameter, J. Nonparametr. Statist., 4, 293-308 (1994) · Zbl 1381.62113
[16] Jan, S.-L.; Randles, R. H., A multivariate signed sum test for the one-sample location problem, J. Nonparametr. Statist., 4, 49-63 (1994) · Zbl 1380.62192
[17] Johnson, N. L.; Kotz, S., Distributions in Statistics: Discrete Distributions (1969), Wiley: Wiley New York · Zbl 0292.62009
[18] Johnson, N. L.; Kotz, S., Distributions in Statistics: Continuous Multivariate Distributions (1972), Wiley: Wiley New York · Zbl 0248.62021
[19] Liu, R. Y.; Singh, K., On a notion of data depth based upon random simplices, J. Amer. Statist. Assoc., 88, 252-260 (1993)
[20] Möttönen, J.; Oja, H.; Tienari, J., On the efficiency of multivariate spatial sign and rank tests, Ann. Statist., 25, 542-552 (1997) · Zbl 0873.62048
[21] Oja, H., Descriptive statistics for multivariate distributions, Statist. Probab. Lett., 1, 327-332 (1983) · Zbl 0517.62051
[22] Oja, H., Affine invariant multivariate sign and rank tests and corresponding estimates: A review, Scand. J. Statist. (1998)
[23] Peters, D.; Randles, R. H., A multivariate signed-rank test for the one-sample location problem, J. Amer. Statist. Assoc., 85, 552-557 (1990) · Zbl 0709.62051
[24] Randles, R. H., A two-sample extension of the multivariate interdirection test, (Dodge, Y., \(L_1 (1992)\), North-Holland: North-Holland Amsterdam), 295-302
[25] Randles, R. H.; Peters, D., Multivariate rank tests for the two-sample location problem, Comm. Statist. Theory Methods, 15, 4225-4238 (1990)
[26] Sneddon, I. N., Elements of partial differential equations (1957), McGraw-Hill: McGraw-Hill New York · Zbl 0077.09201
[27] Tiku, M., Noncentral chi-square distribution, (Kotz, S.; Johnson, N. L.; Read, C. P., Encyclopedia of Statistical Sciences (1985), Wiley: Wiley New York), 276-280
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