Performs the test of multivariate normality of Cox and Small (1978).
test.CS(data, MC.rep = 1000, alpha = 0.05, Points = NULL)a n x d matrix of d dimensional data vectors.
number of repetitions for the Monte Carlo simulation of the critical value.
level of significance of the test.
number of points to approximate the maximum functional on the unit sphere.
a list containing the value of the test statistic, the approximated critical value and a test decision on the significance level alpha:
$Testname of the test.
$Test.valuethe value of the test statistic.
$cvthe approximated critical value.
$Decisionthe comparison of the critical value and the value of the test statistic.
The test statistic is \(T_{n,CS}=\max_{b\in\{x\in\mathbf{R}^d:\|x\|=1\}}\eta_n^2(b)\), where $$\eta_n^2(b)=\frac{\left\|n^{-1}\sum_{j=1}^nY_{n,j}(b^\top Y_{n,j})^2\right\|^2-\left(n^{-1}\sum_{j=1}^n(b^\top Y_{n,j})^3\right)^2}{n^{-1}\sum_{j=1}^n(b^\top Y_{n,j})^4-1-\left(n^{-1}\sum_{j=1}^n(b^\top Y_{n,j})^3\right)^2}$$. Here, \(Y_{n,j}=S_n^{-1/2}(X_j-\overline{X}_n)\), \(j=1,\ldots,n\), are the scaled residuals, \(\overline{X}_n\) is the sample mean and \(S_n\) is the sample covariance matrix of the random vectors \(X_1,\ldots,X_n\). To ensure that the computation works properly \(n \ge d+1\) is needed. If that is not the case the test returns an error. Note that the maximum functional has to be approximated by a discrete version, for details see Ebner (2012).
Cox, D.R., Small, N.J.H. (1978), Testing multivariate normality, Biometrika, 65:263-272.
Ebner, B. (2012), Asymptotic theory for the test for multivariate normality by Cox and Small, Journal of Multivariate Analysis, 111:368-379.
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test.CS(MASS::mvrnorm(10,c(0,1),diag(1,2)),MC.rep=100)
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