Abstract
Changes in the joint distribution of influence functions for the mean vector and the covariance matrix are examined when the true probability distribution is contaminated. In particular, the formulas for influence functions of the first and second moments with respect to the above joint distribution are obtained and used to derive reasonable test statistics for multivariate normality. The formulas are extended by using the joint distribution of score functions for population parameters. An application of the extended formulas to the usual linear regression analysis leads to a measure of multivariate skewness which can be used to reduce the effect of non-normality of the response variable. Also, some relationship between the extended formulas and goodness-of-fit statistics is discussed and used to derive test statistics for multivariate normality.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Box, G. E. P. and Watson, G. S. (1962). Robustness to non-normality of regression test, Biometrika, 49, 93–106.
Cox, D. R. and Small, N. J. H. (1978). Testing multivariate normality, Biometrika, 65, 263–272.
Healy, M. J. R. (1968). Multivariate normal plotting, Appl. Statist. 17, 157–161.
Isogai, T. (1983a). On measures of multivariate skewness and kurtosis, Math. Japon., 28, 251–261.
Isogai, T. (1983b). Monte Carlo study on some measures for evaluating multinormality, Rep. Statist. Appl. Res. Un. Japan. Sci. Engrs., 30, 1–10.
Malkovich, J. F. and Afifi, A. A. (1973). On tests for multivariate normality, J. Amer. Statist. Assoc., 68, 176–179.
Mardia, K. V. (1970). Measures of multivariate skewness and kurtosis with applications, Biometrika, 57, 519–530.
Radhakrishnan, R. and Kshirsagar, A. M. (1981). Influence functions for certain parameters in multivariate analysis, Comm. Statist. A — Theory Methods, 10, 515–529.
Rao, C. R. (1973). Linear Statistical Inference and Its Applications, Wiley, New York.
Spiegelhalter, D. J. (1983). Diagnostic tests of distributional shape, Biometrika, 70, 401–409.
Stuart, A. and Ord, J. K. (1987). Kendall's Advanced Theory of Statistics, Vol. 1, 5th ed., Griffin, London.
Wilks, S. S. (1963). Multivariate statistical outliers, Sankhyā Ser. A, 25, 407–426.
Author information
Authors and Affiliations
About this article
Cite this article
Isogai, T. On using influence functions for testing multivariate normality. Ann Inst Stat Math 41, 169–186 (1989). https://doi.org/10.1007/BF00049116
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00049116