We wish to formulate a test for the hypothesis X/sub i//spl sim/N(/spl mu/,/spl sigma//sup 2/) for i=0,1,...,N-1 against unspecified alternatives. We assume independence of the components of X=[X/sub 0/,X/sub 1/,...,X/sub N-1/]. This is a problem of universal importance as the assumption of Gaussianity is prevalent and fundamental to many statistical theories and engineering applications. Many such tests exist, the most well-known being the /spl chi//sup 2/ goodness-of-fit test with its variants and the Kolmogorov-Smirnov one-sample cumulative probability function test. More powerful modern tests for the hypothesis of Gaussianity include the D'Agostino (1977) K/sup 2/ and Shapiro-Wilk (1968) W tests. Tests for Gaussianity have been proposed which use the characteristic function. It is the purpose of this paper to highlight and resolve problems with these tests and to improve performance so that the test is competitive with, and in some cases better than, the most powerful known tests for Gaussianity.
[1]
S. Shapiro,et al.
A Comparative Study of Various Tests for Normality
,
1968
.
[2]
Ioannis A. Koutrouvelis,et al.
A goodness-of-fit test of simple hypotheses based on the empirical characteristic function
,
1980
.
[3]
É. Moulines,et al.
Testing that a stationary time-series is Gaussian: time-domain vs. frequency-domain approaches
,
1993,
[1993 Proceedings] IEEE Signal Processing Workshop on Higher-Order Statistics.
[4]
T. W. Epps.
Testing That a Stationary Time Series is Gaussian
,
1987
.
[5]
Eric Moulines,et al.
Testing that a multivariate stationary time-series is Gaussian
,
1992,
[1992] IEEE Sixth SP Workshop on Statistical Signal and Array Processing.
[6]
Boualem Boashash,et al.
Estimation of the Characteristic Function
,
1995
.
[7]
E. S. Pearson,et al.
Tests for departure from normality: Comparison of powers
,
1977
.