Solving one-dimensional Stein equations with kernels

Oxford, 2026 — Ferdinand Rapin and Yvik Swan

References

  • Daly, F. A. (2008). Upper bounds for Stein-type operators. Electronic Journal of Probability 13, 566–587. journal
  • Goldstein, L. and Reinert, G. (2013). Stein's method for the Beta distribution and the Pólya–Eggenberger urn. Journal of Applied Probability 50(4), 1187–1205. arXiv
  • Döbler, C. (2015). Stein's method of exchangeable pairs for the Beta distribution and generalizations. Electronic Journal of Probability 20, no. 109, 1–34. arXiv journal
  • Gaunt, R. E., Pickett, A. M. and Reinert, G. (2017). Chi-square approximation by Stein's method with application to Pearson's statistic. Annals of Applied Probability 27(2), 720–756. arXiv journal
  • Döbler, C., Gaunt, R. E. and Vollmer, S. J. (2017). An iterative technique for bounding derivatives of solutions of Stein equations. Electronic Journal of Probability 22, no. 96, 1–39. arXiv journal
  • Ernst, M. and Swan, Y. (2022). Distances between distributions via Stein's method. Journal of Theoretical Probability 35, 949–987. arXiv journal
  • Germain, G. and Swan, Y. (2025). Bounding the \(L^1\)-distance between one-dimensional continuous and discrete distributions via Stein's method. Journal of Theoretical Probability 38(1), 1–43. arXiv journal
  • Gaunt, R. E. (2025). On Stein factors in Stein's method for normal approximation. Statistics & Probability Letters 219, Article 110339. arXiv journal

Some animations in the Gaussian case

For \[ h_z(x)=\mathbf{1}_{(-\infty,z]}(x), \] the standard normal Stein equation \[ f'(x)-xf(x)=h_z(x)-\Phi(z) \] has solution

\[ f_z(x) = \frac{\Phi(x \wedge z)\,\overline{\Phi}(x \vee z)}{\varphi(x)}. \]

The animations below show the evolution of the solution and its first two derivatives as the threshold parameter \(z\) moves from \(-5\) to \(5\).

Evolution of f_z(x) as z moves from -5 to 5
Evolution of \(f_z(x)\) as \(z\) moves from \(-5\) to \(5\).
Evolution of |f_z'(x)| as z moves from -5 to 5
Evolution of \(|f_z'(x)|\) as \(z\) moves from \(-5\) to \(5\).
Evolution of |f_z''(x)| as z moves from -5 to 5
Evolution of \(|f_z''(x)|\) as \(z\) moves from \(-5\) to \(5\).

Interactive beta envelope

Move α and β to explore the beta envelope \[ x \longmapsto \frac{P(x)\overline P(x)}{x(1-x)p(x)}, \qquad Z\sim \mathrm{Beta}(\alpha,\beta). \]

\[ p(x)=\frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}, \qquad P(x)=\int_0^x p(u)\,du, \qquad \overline P(x)=1-P(x). \]

\(x \mapsto P(x)\overline P(x)/(x(1-x)p(x))\).