The Limit Distribution of an Invariant Test Statistic for Multivariate Normality

- Journal title : Communications for Statistical Applications and Methods
- Volume 12, Issue 1, 2005, pp.71-86
- Publisher : The Korean Statistical Society
- DOI : 10.5351/CKSS.2005.12.1.071

Title & Authors

The Limit Distribution of an Invariant Test Statistic for Multivariate Normality

Kim Namhyun;

Kim Namhyun;

Abstract

Testing for normality has always been an important part of statistical methodology. In this paper a test statistic for multivariate normality is proposed. The underlying idea is to investigate all the possible linear combinations that reduce to the standard normal distribution under the null hypothesis and compare the order statistics of them with the theoretical normal quantiles. The suggested statistic is invariant with respect to nonsingular matrix multiplication and vector addition. We show that the limit distribution of an approximation to the suggested statistic is representable as the supremum over an index set of the integral of a suitable Gaussian process.

Keywords

multivariate normality;goodness-of-fit tests;Gaussian process;Brownian bridge;quantile process;

Language

English

Cited by

References

1.

Adler, R. J. (1990). An introduction to continuity, extrema, and related topics for general Gaussian processes, Lecture notes 12, Institute of Mathematical Statistics

2.

Baringhaus, L., and Henze, N. (1988). A consistent test for multivariate normality based on the empirical characteristic function, Metrika, Vol. 35, 339-348

3.

Baringhaus, L., and Henze, N. (1992). Limit distribution for Mardia's measure of multivariate skewness, The Annals of Statistics, Vol. 20, 1889-1902

4.

Billingsley, P. (968). Convergence of Probability Measures, John Wiley, New York

5.

Csorg o, M. (1983). Quantile processes with statistical applications, CBMS-NSF regional conference series in applied mathematics

6.

7.

Csorg o, M., and Revesz, P. (1981). Strong approximations in probability and statistics, Academic Press, New York

8.

D' Agositno, R. B., and Stephens, M. A. (1986). Goodness-of-fit Techniques, Marcel Dekker, New York

9.

del Barrio, E., Cuesta, J. A., Matran, C., and Rodriguez, J. M. (1999). Tests of goodness of fit based on the $L_2-Wasserstein$ distance, The Annals oj Statistics, Vol. 27, 1230-1239

10.

de Wet, T., and Venter, J. H. (1972). Asymptotic distributions of certain test criteria of normality, South African Statistical Journal, Vol. 6, 135-149

11.

de Wet, T., Venter, J. H., and van Wyk, J. W. J. (1979). The null distributions of some test criteria of multivariate normality, South African Statistical Journal, Vol. 13, 153-176

12.

Epps, T. W., and Pulley, I. B. (1983). A test for normality based on the empirical characteristic function, Biometrika, Vol. 70, 723-726

13.

Fattorini, L. (1986). Remarks on the use of the Shapiro-Wilk statistic for testing multivariate normality, Statistica, Vol. 46, 209-217

14.

Finney, R. L., and Thomas, G. B, Jr. (1994). Calculus, Addison-Wesley

15.

Henze, N. (2002). Invariant tests for multivariate normality: A critical review, Statistical Papers, Vol. 43, 467-506

16.

Henze, N., and Wagner, T. (1997). A new approach to the BHEP tests for multivariate normality, Journal of Multivariate Analysis, Vol. 62, 1-23

17.

Henze, N., and Zirkler, H. (1990). A class of invariant and consistent tests for multivariate normality, Communications in Statistics -Theory and Methods, Vol. 19, 3539-3617

18.

Horswell, R. L., and Looney, S. W. (1992). A comparison of tests for multivariate normality that are based on measures of multivariate skewness and kurtosis, Journal of Statistical Computation and Simulation, Vol. 42, 21-38

19.

Kim, N. (1994). Goodness of fit tests for bivariate distributions, Ph. D. dissertation, University of California, Berkeley

20.

Kim, N. (2004). An approximate Shapiro-Wilk statistic for testing multivariate normality, The Korean Journal of Applied statistics, Vol. 17, 35-37

21.

Kim, N., and Bickel, P. J (2003). The limit distribution of a test statistic for bivariate normality, Statistica Sinica, Vol. 13, 327-349

22.

Leslie, J. R., Stephens, M. A., and Fotopolous, S. (1986). Asymptotic distribution of the Shapiro-Wilk W for testing for normality, The Annals of Statistics, Vol. 14, 1497-1506

23.

Liang, J., Li, R., Fang, H., and Fang, K.-T. (2000). Testing multinormality based on low-dimensional projection, Journal Statistical planning and Inference, Vol. 86, 129-141

24.

Machado, S. G. (1983). Two statistics for testing for multivariate normality, Biometrika, Vol. 70, 713-718

25.

Malkovich, J. F., and Afifi, A. A. (1973). On tests for multivariate normality, Journal of the American Statistical Association, Vol. 68, 176-179

26.

Mardia, K. V. (1970). Measures of multivariate skewness and kurtosis with applications, Biometrika, Vol. 57, 519-530

27.

Mardia, K. V. (1974). Applications of some measures of multivariate skewness and kurtosis for testing normality and robustness studies, Sankhya A, Vol. 36, 115-128

28.

Mardia, K. V. (1975). Assessment of multinormality and the robustness of Hotelling's $T^2$ test, Applied Statistics, Vol. 24, 163-171

29.

Massart, P. (1989). Strong approximation for multivarite empirical and related processes, via KMT constructions, The Annals of probability, Vol. 17, 266-291

30.

Romeu, J. L., and Ozturk, A. (1993). A comparative study of goodness-of-fit tests for multivariate normality, Journal of Multivariate analysis, Vol. 46, 309-334

31.

Roy, S. N. (1953). On a heuristic method of test construction and its use in multivariate analysis, Annals of Mathematical Statistics, Vol. 24, 220-238

32.

Shapiro, S. S., and Wilk, M. B. (1965). An analysis of variance test for normality (complete samples), Biometrika, Vol. 52, 591-611

33.

Thode, Jr. H. C. (2002). Testing for Normality. Marcel Dekker, New York