F-probability plot and its application to multivariate normality

We introduce the squared jackknife distance of a multivariate observation and derive its finite sampling distribution when a random sample is taken from a multivariate normal distribution. Based on the finite sampling distribution of the squared jackknife distance we recommend the use of an F-probability plot for checking multivariate normality and the F-probability plot correlation coefficient and the F-probability plot intercept as numerical measures of detecting deviation from normality. We report empirical percentiles of the F-probability plot correlation coefficient and the F-probability plot intercept. We compare the powers of tests for multivariate normality based on the F-probability plot and based on the “usual” chi-square probability plot and conduct extensive power studies of the proposed numerical measures.

[1]  R. Tibshirani,et al.  An Introduction to the Bootstrap , 1995 .

[2]  R. McCulloch,et al.  A Multivariate Generalization of Quantile-Quantile Plots , 1990 .

[3]  J. Leslie,et al.  Asymptotic Distribution of the Shapiro-Wilk $W$ for Testing for Normality , 1986 .

[4]  S. Looney,et al.  Use of the Correlation Coefficient with Normal Probability Plots , 1985 .

[5]  H. Oja.  Descriptive Statistics for Multivariate Distributions , 1983 .

[6]  David R. Cox,et al.  Testing multivariate normality , 1978 .

[7]  Ramanathan Gnanadesikan,et al.  Methods for statistical data analysis of multivariate observations , 1977, A Wiley publication in applied statistics.

[8]  K. V. Mardia,et al.  Algorithm AS 84: Measures of Multivariate Skewness and Kurtosis , 1975 .

[9]  K. Mardia.  Assessment of multinormality and the robustness of Hotelling's T^2 test , 1975 .

[10]  J. Filliben.  The Probability Plot Correlation Coefficient Test for Normality , 1975 .

[11]  S. Shapiro,et al.  An Approximate Analysis of Variance Test for Normality , 1972 .

[12]  R. D'Agostino.  An omnibus test of normality for moderate and large size samples , 1971 .

[13]  K. Mardia.  Measures of multivariate skewness and kurtosis with applications , 1970 .

[14]  G. Zyskind.  Introduction to Matrices with Applications in Statistics , 1970 .

[15]  M. Healy,et al.  Multivariate Normal Plotting , 1968 .

[16]  S. Shapiro,et al.  An Analysis of Variance Test for Normality (Complete Samples) , 1965 .

[17]  T. W. Anderson,et al.  An Introduction to Multivariate Statistical Analysis , 1959 .

[18]  L. Weiss.  A Test of Fit for Multivariate Distributions , 1958 .

[19]  Maurice G. Kendall,et al.  The advanced theory of statistics , 1945 .

[20]  E. S. Pearson.  A FURTHER DEVELOPMENT OF TESTS FOR NORMALITY , 1930 .

[21]  A. Madansky.  Prescriptions for working statisticians , 1988 .

[22]  J. Overall,et al.  Applied multivariate analysis , 1983 .