Poisson kernel-based quadratic distance test of Uniformity on the sphere

Poisson kernel-based quadratic distance test of Uniformity on the sphere#

Let \(x_1, x_2, \ldots, x_n\) be a random sample with empirical distribution function \(\hat F\). We test the null hypothesis of uniformity on the \((d-1)\)-dimensional sphere, i.e., \(H_0: F = G\), where \(G\) is the uniform distribution on the \((d-1)\)-dimensional sphere \(\mathcal{S}^{d-1}\).

We compute the U-statistic estimate of the sample KBQD (Kernel-Based Quadratic Distance):

\[U_{n} = \frac{1}{n(n-1)} \sum_{i=2}^{n} \sum_{j=1}^{i-1} K_{cen} (\mathbf{x}_{i}, \mathbf{x}_{j}),\]

then the first test statistic is given as:

\[T_{n} = \frac{U_{n}}{\sqrt{Var(U_{n})}},\]

with:

\[Var(U_{n}) = \frac{2}{n(n-1)} \left[\frac{1+\rho^{2}}{(1-\rho^{2})^{d-1}} - 1\right],\]

and the V-statistic estimate of the KBQD:

\[V_{n} = \frac{1}{n} \sum_{i=1}^{n} \sum_{j=1}^{n} K_{cen} (\mathbf{x}_{i}, \mathbf{x}_{j}),\]

where \(K_{cen}\) denotes the Poisson kernel \(K_\rho\) centered with respect to the uniform distribution on the \((d-1)\)-dimensional sphere, that is:

\[K_{cen}(\mathbf{u}, \mathbf{v}) = K_\rho(\mathbf{u}, \mathbf{v}) - 1\]

and:

\[K_\rho(\mathbf{u}, \mathbf{v}) = \frac{1-\rho^{2}}{\left(1+\rho^{2}- 2\rho (\mathbf{u} \cdot \mathbf{v})\right)^{d/2}},\]

for every \(\mathbf{u}, \mathbf{v} \in \mathcal{S}^{d-1} \times \mathcal{S}^{d-1}\).

The asymptotic distribution of the V-statistic is an infinite combination of weighted independent chi-squared random variables with one degree of freedom. The cutoff value is obtained using the Satterthwaite approximation \(c \cdot \chi_{DOF}^2\), where:

\[c = \frac{(1+\rho^{2}) - (1-\rho^{2})^{d-1}}{(1+\rho)^{d} - (1-\rho^{2})^{d-1}}\]

and:

\[DOF(K_{cen}) = \left(\frac{1+\rho}{1-\rho}\right)^{d-1}\left\{ \frac{\left(1+\rho - (1-\rho)^{d-1}\right)^{2}} {1+\rho^{2} - (1-\rho^{2})^{d-1}}\right\}.\]

For the \(U\)-statistic, the cutoff is determined empirically:

  • Generate data from a Uniform distribution on the \(d\)-dimensional sphere;

  • Compute the test statistics for num_iter Monte Carlo (MC) replications;

  • Compute the 95th quantile of the empirical distribution of the test statistic.