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Locally most powerful rank tests of independence for copula models. (English) Zbl 1065.62081
Summary: A formula is given for a statistic that provides a locally most powerful rank test of independence against alternatives expressed by copula models. The Savage, Spearman and van der Waerden statistics are seen to be optimal in special cases of interest. The asymptotic relative efficiency (ARE) of any linear rank procedure with respect to the optimal test is expressed as a squared correlation in which the bivariate dependence structure of the data only enters through the copulas. In contrast, the margins are shown to influence the ARE of the best rank test, compared to the standard test of independence based on Pearson’s correlation. An extensive simulation study is used to assess the effect of both the margins and the dependence structure on the power of several parametric and nonparametric procedures in small samples from a variety of bivariate distributions.

MSC:
62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
62E20 Asymptotic distribution theory in statistics
65C60 Computational problems in statistics (MSC2010)
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References:
[1] Frees E. W., North American Actuarial Journal 2 pp 1– (1998) · Zbl 1081.62564 · doi:10.1080/10920277.1998.10595667
[2] Embrechts P., United Kingdom:, in: Risk Management: Value at Risk and Beyond pp 176– (2002) · doi:10.1017/CBO9780511615337.008
[3] Mosler K., Stochastic Orders and Applications: A Classified Bibliography (1993) · Zbl 0805.62001 · doi:10.1007/978-3-642-49972-2
[4] Joe H., Multivariate Models and Dependence Concepts (1997) · Zbl 0990.62517 · doi:10.1201/b13150
[5] Nelsen R. B., An Introduction to Copulas, Lecture Notes in Statistics No. 139 (1999) · Zbl 0909.62052 · doi:10.1007/978-1-4757-3076-0
[6] Drouet-Mari D., Correlation and Dependence (2001) · Zbl 0977.62004
[7] Bhuchongkul S., The Annals of Mathematical Statistics 35 pp 138– (1964) · Zbl 0126.15003 · doi:10.1214/aoms/1177703735
[8] Behnen K., The Annals of Mathematical Statistics 42 pp 325– (1971) · Zbl 0224.62019 · doi:10.1214/aoms/1177693515
[9] Behnen K., The Annals of Mathematical Statistics 43 pp 1839– (1972) · Zbl 0255.62042 · doi:10.1214/aoms/1177690855
[10] Shirahata S., Bulletin of Mathematical Statistics 16 (1974)
[11] Shirahata S., The Annals of Statistics 3 pp 241– (1975) · Zbl 0303.62041 · doi:10.1214/aos/1176343014
[12] Ciesielska L., Zastosowania Matematyki 18 pp 61– (1983)
[13] Stuart A., Skandinavisk Aktuarietidskrift 37 pp 163– (1954)
[14] Konijn H. S., The Annals of Mathematical Statistics 27 pp 300– (1956) · Zbl 0075.29302 · doi:10.1214/aoms/1177728260
[15] Gokhale D. V., Annals of the Institute of Statistical Mathematics 20 pp 255– (1968) · Zbl 0165.21501 · doi:10.1007/BF02911639
[16] Hájek J., Theory of Rank Tests (1967) · Zbl 0161.38102
[17] Garralda-Guillem A. I., Dependencia y tests de rangos para leyes bidimensionales (1997)
[18] Plackett R. L., Journal of the American Statistical Association 60 pp 516– (1965) · doi:10.1080/01621459.1965.10480807
[19] Ali M. M., Journal of Multivariate Analysis 8 pp 405– (1978) · Zbl 0387.62019 · doi:10.1016/0047-259X(78)90063-5
[20] Frank M. J., Aequationes Mathe-maticae 19 pp 194– (1979) · Zbl 0444.39003 · doi:10.1007/BF02189866
[21] Oakes D., Biometrika 90 pp 478– (2003) · Zbl 1034.62050 · doi:10.1093/biomet/90.2.478
[22] Genest C., The Canadian Journal of Statistics 14 pp 145– (1986) · Zbl 0605.62049 · doi:10.2307/3314660
[23] Puri M. L., Nonparametric Methods in Multivariate Analysis (1971) · Zbl 0237.62033
[24] Clayton D. G., Biometrika 65 pp 141– (1978)
[25] David H. A., Order Statistics, 2. ed. (1982)
[26] Gumbel E. J., Journal of the American Statistical Association 55 pp 698– (1960) · doi:10.1080/01621459.1960.10483368
[27] Barnett V., Communications in Statistics, Theory and Methods 9 pp 453– (1980) · Zbl 0449.62011 · doi:10.1080/03610928008827893
[28] Verret F., Unpublished Master’s thesis No 21063, in: Tests de rangs localement les plus puissants pour l’hypothèse d’indépendance bivariée
[29] van der Vaart A. W., Asymptotic Statistics (1998) · Zbl 0910.62001 · doi:10.1017/CBO9780511802256
[30] van Eeden C., The Annals of Mathematical Statistics 34 pp 1442– (1963) · Zbl 0128.38505 · doi:10.1214/aoms/1177703876
[31] Genest C., Biometrika 82 pp 543– (1995) · Zbl 0831.62030 · doi:10.1093/biomet/82.3.543
[32] Blest D. C., Australian and New Zealand Journal of Statistics 42 pp 101– (2000) · Zbl 0977.62061 · doi:10.1111/1467-842X.00110
[33] Genest C., The Canadian Journal of Statistics 31 pp 35– (2003) · Zbl 1035.62058 · doi:10.2307/3315902
[34] Cooke R. M., Risk Analysis 6 pp 335– (1986) · doi:10.1111/j.1539-6924.1986.tb00226.x
[35] Ferguson T. S., Statistical Papers 36 pp 41– (1995) · Zbl 0817.62040 · doi:10.1007/BF02926016
[36] Gieser P. W., Journal of the American Statistical Association 92 pp 561– (1997) · doi:10.1080/01621459.1997.10474008
[37] Cuzick J., Biometrika 69 pp 351– (1982) · Zbl 0497.62039 · doi:10.1093/biomet/69.2.351
[38] Shih J. H., Biometrics 52 pp 1440– (1996) · Zbl 0897.62123 · doi:10.2307/2532857
[39] Säid M., CA: 42, in: Mathematical Statistics and Applications: Festschrift for Constance van Eeden, IMS Lecture Notes, Monograph Series pp 435– (2003)
[40] Kallenberg W. C.M., Journal of the American Statistical Association 94 pp 285– (1999) · doi:10.1080/01621459.1999.10473844
[41] Hallin M., New Directions in Time Series Analysis pp 112– (1992)
[42] Hoeffding W., The Annals of Mathematical Statistics 19 pp 546– (1948) · Zbl 0032.42001 · doi:10.1214/aoms/1177730150
[43] Blum J. R., The Annals of Mathematical Statistics 32 pp 485– (1961) · Zbl 0139.36301 · doi:10.1214/aoms/1177705055
[44] Genest C., Test 13 pp 335– (2004) · Zbl 1069.62039 · doi:10.1007/BF02595777
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