This function gives a test on whether the data is elliptically distributed based on kernel embedding of probability. See Tang and Li (2024) for details. Gaussian kernels and product-type inverse quadratic kernels are considered.
EllKEPT(
X,
eps = 1e-06,
kerU = "Gaussian",
kerTheta = "Gaussian",
gamma.U = 0,
gamma.Theta = 0
)A list of the following:
The value of the test statistic.
The p-value of the test.
The n eigenvalues in the approximated asymptotic
distribution.
The tuning parameter gamma.U used in the test. Same as
the input if its input is nonzero.
The tuning parameter gamma.Theta used in the test.
Same as the input if its input is nonzero.
A matrix with n rows and d columns.
The regularization constant added to the diagonal to avoid
singularity. Default value is 1e-6.
The type of kernel function on U. Currently supported
options are "Gaussian" and "PIQ".
The type of kernel function on Theta. Currently
supported options are "Gaussian" and "PIQ".
The tuning parameter gamma in the kernel function
k_U(u1,u2). If gamma.U=0, the recommended procedure of selecting
tuning parameter will be applied. Otherwise, the value given in
gamma.U will be directly used as the tuning parameter. Default value
is gamma.U=0. See "Details" for more information.
The tuning parameter gamma in the kernel function
k_Theta(theta1,theta2). If gamma.Theta=0, the recommended procedure
of selecting tuning parameter will be applied. Otherwise, the value given in
gamma.Theta will be directly used as the tuning parameter. Default
value is gamma.Theta=0. See "Details" for more information.
The Gaussian kernel is defined as k(z1,z2)=exp(-gamma*||z1-z2||^2), and the
Product-type Inverse-Quadratic (PIQ) kernel is defines as
k(z1,z2)=Prod_j(1/(1+gamma*(z1_j-z2_j)^2)). The recommended procedure of
selecting tuning parameter is given as in the simulation section of Tang and
Li (2023+), where we set
1/sqrt(gamma)=(n(n-1)/2)^(-1)*sum_{1<=i<j<=n}||Z_i-Z_j||.
Tang, Y. and Li, B. (2024), “A nonparametric test for elliptical distribution based on kernel embedding of probabilities,” https://arxiv.org/abs/2306.10594
set.seed(313)
n=50
d=3
## Null Hypothesis
X=matrix(rnorm(n*d),nrow=n,ncol=d)
EllKEPT(X)
## Alternative Hypothesis
X=matrix(rchisq(n*d,2)-2,nrow=n,ncol=d)
EllKEPT(X)
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