EllKEPT {KEPTED}

Kernel embedding of probability test for elliptical distribution

Description

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.

Usage

EllKEPT(
  X,
  eps = 1e-06,
  kerU = "Gaussian",
  kerTheta = "Gaussian",
  gamma.U = 0,
  gamma.Theta = 0
)

Arguments

X

A matrix with n rows and d columns.

eps

The regularization constant added to the diagonal to avoid singularity. Default value is 1e-6.

kerU

The type of kernel function on U. Currently supported options are "Gaussian" and "PIQ".

kerTheta

The type of kernel function on Theta. Currently supported options are "Gaussian" and "PIQ".

gamma.U

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.

gamma.Theta

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.

Details

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||.

Value

A list of the following:

stat

The value of the test statistic.

pval

The p-value of the test.

lambda

The n eigenvalues in the approximated asymptotic distribution.

gamma.U

The tuning parameter gamma.U used in the test. Same as the input if its input is nonzero.

gamma.Theta

The tuning parameter gamma.Theta used in the test. Same as the input if its input is nonzero.

Note

In the arguments, eps refers to a regularization constant added to the diagonal. When the dimension is high, we recommend increasing eps to avoid singularity.

References

Tang, Y. and Li, B. (2024), “A nonparametric test for elliptical distribution based on kernel embedding of probabilities,” https://arxiv.org/abs/2306.10594

Examples

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)