Goodness-of-fit tests for cone-sample uniformity

Performs Kolmogorov-Smirnov and chi-squared goodness-of-fit tests to check whether a sample matrix produced by generate_cone_sample or generate_cone_samples is consistent with the uniform distribution on the spherical cap of axis \(\hat{\mu}\) and half-angle \(\theta_0\).

Usage

verify_cone_uniformity(samples, mu_hat, theta0, n_bins = 20)

Arguments

samples

Matrix. Each column is a sample point (from generate_cone_samples).

mu_hat

Numeric vector. Central axis direction.

theta0

Numeric. Maximum planar angle of the cone.

n_bins

Integer. Number of bins for histogram (default: 20).

Value

List containing:

ks_statistic

Kolmogorov-Smirnov test statistic

ks_pvalue

p-value for the KS test

chi_squared

Chi-squared goodness-of-fit statistic

chi_pvalue

p-value for chi-squared test

angles

Vector of angles between samples and \(\hat{\mu}\)

Details

Verify uniformity of cone sampling

For each sample column, the function computes the angle \(\theta\) to the central axis \(\hat{\mu}\). It then runs a Kolmogorov-Smirnov test comparing the empirical CDF of \(\theta\) to the theoretical \(F_\theta(\theta) = \Theta(\theta) / \Theta(\theta_0)\), and a chi-squared goodness-of-fit test on n_bins angular bins. Bins with expected count below 5 are dropped before the chi-squared aggregation. If ties in the sampled angles are detected (a finite- precision artefact), a small jitter is applied before the KS test to avoid spurious warnings.

References

Arun, I., & Venkatapathi, M. (2025). An O(n) algorithm for generating uniform random vectors in n-dimensional cones. Sankhya A, 87(2), 327-348. doi:10.1007/s13171-025-00387-9

See also

generate_cone_sample, generate_cone_samples, generate_hollow_cone_sample for the samplers under test; theta_to_omega for the underlying CDF; rejection_cost for the expected cost of the rejection planar-angle generator.

Examples

if (FALSE) { # \dontrun{
samples <- generate_cone_samples(10000, c(0, 0, 1), pi / 4)
fit <- verify_cone_uniformity(samples, c(0, 0, 1), pi / 4)
c(KS_p = fit$ks_pvalue, chi2_p = fit$chi_pvalue)
} # }