The multivariate sample distributions considered to this end are convex or radially concave contoured.
A brief comparison of norms and antinorms, generating convex or radially concave level sets of the multivariate sample density, respectively, is given in the Appendix.
The most general non-standard distributions of the multivariate sample vector we are considering in the present paper are star-shaped distributions.
Exact distributions of generalized Chi-square and Fisher statistics are used to derive confidence intervals and significance tests for inferring on one or two scaling parameters, respectively, under non-standard assumptions w.r.t. the multivariate sample distribution.
Since the Sobol method uses a pseudo-randomized multivariate sampling procedure, it can be used as a basis for a global uncertainty evaluation by constructing the pdfs and cumulative distribution functions for each of the selected outputs.
A multivariate proportional sampling strategy within each stratum was determined to be the optimal procedure for identifying a sample of accessions that best represents the diversity of the total collection [ 46].
That the exact null distribution of A1 is known, regardless of the underlying data distribution, is a particularly attractive property for a multivariate two-sample test.
To overcome these difficulties, we used an elegant algorithm proposed in [ 55] which is based on a multivariate Boltzmann sampling scheme, initially introduced in the context of enumerative combinatorics.
Instead, in VSEAMS, we adapt a multivariate normal sampling approach, which we show is not only faster than phenotype permutation, but can be applied in the typical case where raw genotype data are not available.
Gene-sets with nominally significant burden for at least one variant category (P < 0.10 for LoF and splicing regulatory, and P < 0.05 for missense variants) were then tested for the joint burden of the three variant categories with a multivariate, two-sample Hotelling's T-Square test (Hotelling 1931).
