Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.32506

Direct Simulation of Poisson and Newton Kernel Distributions

Lukas Sablica·Kurt Hornik
View original source ↗Read source document ↗Export metadata ↓

Abstract

The Poisson kernel-based (PKB) distribution is a flexible and numerically convenient family on the unit sphere used for model-based clustering, uniformity testing, and directional data analysis. Available random variate generators rely on rejection from a von Mises-Fisher, angular central Gaussian, or projected Saw envelope and therefore require a rejection loop, tuning or root-finding, or a univariate setup phase. We show that none is needed. A uniformly random line through the interior point $ρμ$ meets the sphere in two points; selecting one with probability proportional to the opposite segment length returns exactly the target density. The sampler is exact, rejection- and tuning-free, uses one uniform direction and one Bernoulli draw, and costs $O(d)$ operations. We place the distribution in the power-Saw family of densities proportional to $\|x-ρμ\|^{-ξ}$ and show that, for $d\geq3$, the same construction is exact for one further exponent. This introduces the Newton kernel-based (NKB) distribution, whose normalizing constant is identically one. Across a wide grid of dimensions, concentrations, and sample sizes, the direct sampler is faster than the better of two previously recommended methods in every case, with the largest gains at moderate concentration.

Explore related subjects

stat.COstat.APstat.ME

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Lukas Sablica, Kurt Hornik. 2026-10-23. Direct Simulation of Poisson and Newton Kernel Distributions. https://arxiv.org/abs/2610.32506

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

SVEMnet: An R package for Self-Validated Elastic-Net Ensembles and Multi-Response Optimization in Small-Sample Mixture-Process Experiments

SVEMnet is an R package for fitting Self-Validated Ensemble Models (SVEM) with elastic-net base learners and performing multi-response optimization in small-sample mixture-process design-of-experiments (DOE) studies with numeric, categorical, and mixture factors. SVEMnet wraps elastic-net and relaxed elastic-net models for Gaussian and binomial responses from glmnet in a fractional random-weight (FRW) resampling scheme with anti-correlated train/validation weights; penalties are selected by validation-weighted AIC- and BIC-type criteria, and predictions are averaged across replicates to stabilize fits near the interpolation boundary. In addition to the core SVEM engine, the package provides deterministic high-order formula expansion, a permutation-based whole-model test heuristic, and a mixture-constrained random-search optimizer that combines Derringer-Suich desirability functions, bootstrap-based uncertainty summaries, and optional mean-level specification-limit probabilities to generate scored candidate tables and diverse exploitation and exploration medoids for sequential fit-score-run-refit workflows. A simulated lipid nanoparticle (LNP) formulation study illustrates these tools in a small-sample mixture-process DOE setting, and simulation experiments based on sparse quadratic response surfaces benchmark SVEMnet against repeated cross-validated elastic-net baselines.

stat.CO↗
arXiv2026

A General Framework for Importance Sampling with Markov Random Walks

Although stochastic models driven by latent Markov processes are widely used, the classical importance sampling methods based on the exponential tilting for these models suffer from difficulties in computing associated eigenvalues and eigenfunctions and the plausibility of the indirect asymptotic large deviation regime for the variance of the estimator. We propose an importance sampling framework that twists the observable and latent processes separately using a link function that directly minimizes the estimator's variance. An optimal link function is chosen within the locally asymptotically normal family. We show the logarithmic efficiency of the estimator. As applications, we estimate an overflow probability under a pandemic model and the CoVaR, a measurement of the co-dependent financial systemic risk; both are beyond the scope of traditional importance sampling methods due to nonlinear features.

stat.CO↗
arXiv2026

Uniform ergodicity and spectral gap of importance-tempered MCMC with heavy-tailed targets

We study the importance-tempered random-walk Metropolis (RWM) algorithm on $\mathbb{R}^d$, which targets the tempered density $π_β(x) \propto π(x)^β$ with $β\in (0,1)$ and corrects for the resulting bias by importance weighting. The dynamics of this sampler can be described by a continuous-time Markov chain whose embedded chain is the tempered RWM. We prove that when the tails of $π$ decay at rate $\|x\|^{-γ}$, this continuous-time chain is uniformly ergodic if and only if $d/γ< β< (γ- 2)/γ$. Further, the global spectral gap of this chain can be controlled via the spectral gap of the chain restricted to a ball with radius growing polynomially in $d$. Our theory and simulation studies show that, with a suitable choice of $β$, importance tempering can substantially reduce sensitivity to initialization and significantly improve efficiency over the standard RWM.

stat.CO↗