mnt (version 1.3)

CS: Statistic of the test of Cox and Small

Description

This function returns the (approximated) value of the test statistic of the test of Cox and Small (1978).

Usage

CS(data, Points = NULL)

Arguments

data

a n x d matrix of d dimensional data vectors.

Points

points for approximation of the maximum on the sphere. Points=NULL generates 5000 uniformly distributed Points on the d dimensional unit sphere.

Value

approximation of the value of the test statistic of the test of Cox and Small (1978).

Details

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 function returns an error. Note that the maximum functional has to be approximated by a discrete version, for details see Ebner (2012).

References

Cox, D.R. and Small, N.J.H. (1978), Testing multivariate normality, Biometrika, 65:263<U+2013>272.

Ebner, B. (2012), Asymptotic theory for the test for multivariate normality by Cox and Small, Journal of Multivariate Analysis, 111:368<U+2013>379.

Examples

# NOT RUN {
CS(MASS::mvrnorm(50,c(0,1),diag(1,2)))

# }