×

Goodness-of-fit analysis for multivariate normality based on generalized quantiles. (English) Zbl 1061.62532

Summary: Based on the notion of multivariate quantiles in terms of minimum volume ellipsoids as introduced by Einmahl and Mason, a generalized chi-square quantile plot is constructed from which tests for multivariate normality are derived. The generalized quantile plot also offers some extra possibilities for data analysis, e.g. in identifying and describing non-normality. An algorithm is proposed to approximate the generalized quantiles, extending existing methods to approximate median quantiles with only a minor extra computational cost and preserving the effectivity of the method. A power analysis is included and the method is illustrated with some case-studies.

MSC:

62H15 Hypothesis testing in multivariate analysis
62G10 Nonparametric hypothesis testing
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Atkinson, A. C.: Fast very robust methods for the detection of multiple outliers. J. amer. Statist. assoc. 89, 1329-1339 (1994) · Zbl 0825.62429
[2] Bingham, N.H., Goldie, C.M., Teugels, J.L., 1987. Regular Variation. Cambridge University Press, Cambridge. · Zbl 0617.26001
[3] Beirlant, J.; Vynckier, P.; Teugels, J. L.: Tail index estimation, Pareto quantile plots, and regression diagnostics. J. am. Statist. assoc. 91, 1659-1667 (1996) · Zbl 0881.62077
[4] Cook, R. D.; Hawkins, D. M.; Weisberg, S.: Exact iterative computation of the robust multivariate minimum volume ellipsoid estimator. Statist. probab. Lett. 16, 213-218 (1993)
[5] Cook, R. D.; Johnson, M. E.: Generalized burr-Pareto-logistic distributions with applications to an uranium exploration data set. Technometrics 28, 123-131 (1986)
[6] Easton, G. S.; Mccullogh, R. E.: A multivariate generalization of quantile–quantile plots. J. amer. Statist. assoc. 85, 376-386 (1990)
[7] Einmahl, J. H. J.; Mason, D. M.: Generalized quantile processes. Ann. statist. 20, 1062-1078 (1992) · Zbl 0757.60012
[8] Hadi, A. S.: Identifying multiple observations in multivariate data. J. roy. Statist. soc. Ser. B 54, 761-771 (1992)
[9] Hawkins, D. M.: A new test for multivariate normality and homoscedasticity. Technometrics 23, 105-110 (1981) · Zbl 0466.62044
[10] Hawkins, D. M.: A feasible solution algorithm for the minimum volume ellipsoid estimator in multivariate data. Comput. statist. 8, 95-107 (1993)
[11] Hill, B. M.: A simple approach to inference about the tail of a distribution. Ann. statist. 3, 1163-1174 (1975) · Zbl 0323.62033
[12] Jackson, O.: An analysis of departures from the exponential distribution. J. roy. Statist. soc. Ser. B 29, 540-549 (1967) · Zbl 0183.21402
[13] Koziol, J. A.: A class of invariant procedures for assessing multivariate normality. Biometrika 69, 423-427 (1982) · Zbl 0494.62027
[14] Lariccia, V.; Mason, D. M.: Cramér-von Mises statistics based on the sample quantile function and estimated parameters. J. multivariate statist. 18, 93-106 (1986) · Zbl 0583.62015
[15] Looney, S. W.: How to use test for multivariate normality to assess multivariate normality. Amer. statist. 49, 64-69 (1995)
[16] Mardia, K. V.: Measures of multivariate skewness and kurtosis with applications. Biometrika 57, 519-530 (1970) · Zbl 0214.46302
[17] Moran, P. A. P.: Testing for correlation between non-negative variates. Biometrika 54, 385-394 (1967) · Zbl 0166.15101
[18] Pettit, A.: Generalized cramér-von Mises statistics for the gamma distribution. Biometrika 65, 232-235 (1978) · Zbl 0375.62026
[19] Rocke, D. M.; Woodruff, D. L.: Identification of outliers in multivariate data. J. amer. Statist. assoc. 91, 1047-1061 (1996) · Zbl 0882.62049
[20] Romeu, J. L.; Ozturk, A.: A comparative study of goodness-of-fit tests for multivariate normality. J. multivariate analysis 25, 309-334 (1993) · Zbl 0778.62051
[21] Rousseeuw, P.J., 1985. Multivariate estimation with high breakdown point. In: Grossman, W., Pflug, G., Vincze, I., Wertz, W. (Eds.), Mathematical Statistics and Applications, Reidel, Dordrecht, pp. 283–297. · Zbl 0609.62054
[22] Rousseeuw, P.J., Leroy, A., 1987. Robust Regression and Outlier Detection. Wiley, New York. · Zbl 0711.62030
[23] Rousseeuw, P.J., van Zomeren, B.C., 1990. Unmasking multivariate outliers and leverage points. J. Amer. Statist. Assoc. 85, 633–639.
[24] Rousseeuw, P.J., van Zomeren, B.C., 1991. Robust distances: simulations and cutoff values. In: Stahel, W., Weisberg, S. (Eds.), Directions in Robust Statistics and Diagnostics, Part II, IMA Vol. in Math. Appl. 34, 185–194. Springer, New York.
[25] Shorack, G.R., Wellner, J.A., 1986. Empirical Processes with Applications to Statistics. Wiley, New York. · Zbl 1170.62365
[26] Titterington, D. M.: Estimation of correlation coefficients by ellipsoidal trimming. J. roy. Statist. soc. Ser. C 27, 227-234 (1978) · Zbl 0436.62062
[27] Vynckier, C., 1997. Applications of Generalized Quantiles in Multivariate Statistics. Ph.D. Thesis, Department of Mathematics, Katholieke Universiteit Leuven.
[28] De Wet, T.; Venter, J. H.: A goodness-of-fit test for a scale parameter family of distributions. S. african statist. J. 7, 35-46 (1973) · Zbl 0258.62027
[29] Woodruff, D.; Rocke, D. M.: Heuristic search algorithms for the minimum volume ellipsoid. J. comput. Graphical statist. 2, 69-95 (1993)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.