arXiv 6 Aug 2026 · Mathematics — Statistics Theory
arXiv:2608.06116 · PDF · Extracted main text
The extended-onion and C-vine constructions of Lewandowski, Kurowicka and Joe (2009) are standard methods for sampling from the $LKJ_n(η)$ distribution on correlation matrices. We show that both arise from the simpler row-normalized Bartlett construction associated with the restricted-Wishart representation of Wang, Wu and Chu (2018), which reuses random quantities that the classical samplers regenerate. Two exact row-wise couplings establish this: the squared norm of the Gaussian vector supplying the onion's direction has exactly the Gamma law required for one component of the Beta radius, and the same vector, with one chi-squared variate, generates the entire row of mutually independent C-vine partial correlations with their required symmetric-Beta laws. Under gamma-ratio accounting, normalized Bartlett, the onion, and the conventional symmetric-Beta C-vine require $n-1$, $2(n-1)$, and $n(n-1)$ Gamma-equivalent calls. Controlled benchmarks confirm a low-dimensional advantage over the onion implementation and a persistent advantage over the C-vine implementations examined; direct Bartlett normalization also avoids subtractive complements, moving the small-$η$ zero-diagonal threshold from machine-epsilon scale toward the subnormal range. The sampler is valid for every real $η>0$ and requires only standard normal and chi-squared variates.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Wang, Z., Wu, Y., and Chu, H (2018) On equivalence of the LKJ distribution and the restricted Wishart distribution | 1.000 | 9 | 5 | 100% |
| 2 | Joe, H. and Kurowicka, D (2026) Random correlation matrices generated via partial correlation C-vines | 1.000 | 5 | 3 | 100% |
| 3 | Lewandowski, D., Kurowicka, D., and Joe, H (2009) Generating random correlation matrices based on vines and extended onion method | 0.965 | 10 | 6 | 90% |
| 4 | Carpenter, B., Gelman, A., Hoffman, M. D., Lee, D., Goodrich, B., Be… (2017) Stan: A probabilistic programming language | 0.737 | 3 | 2 | 100% |
| 5 | Besancon, M., Papamarkou, T., Anthoff, D., Arslan, A., Byrne, S., Li… (2021) Distributions.jl: Definition and modeling of probability distributions in the JuliaStats ecosystem | 0.644 | 2 | 2 | 100% |
| 6 | Hansen, P. R (2026) Correlation matrices in high dimensions: The elliptope as a sample-correlation ensemble self | 0.511 | 2 | 2 | 50% |
| 7 | Chen, J. and Revels, J (2016) Robust benchmarking in noisy environments | 0.511 | 2 | 1 | 100% |
| 8 | Barndorff-Nielsen, O. E. and Schou, G (1973) On the parametrization of autoregressive models by partial autocorrelations | 0.405 | 1 | 1 | 100% |
| 9 | Bartlett, M. S (1939) A note on tests of significance in multivariate analysis | 0.405 | 1 | 1 | 100% |
| 10 | Bezanson, J., Edelman, A., Karpinski, S., and Shah, V. B (2017) Julia: A fresh approach to numerical computing | 0.405 | 1 | 1 | 100% |
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