Asymptotic tail properties of the distributions in the class of dispersion models

The class of dispersion models introduced by J{\o}rgensen (1997b) covers many known distributions such as the normal, Student t, gamma, inverse Gaussian, hyperbola, von-Mises, among others. We study the small dispersion asymptotic (J{\o}rgensen, 1987b) behavior of the probability density functions of dispersion models which satisfy the uniformly convergent saddlepoint approximation. Our results extend those obtained by Finner et al. (2008).

💡 Research Summary

The paper investigates the tail behavior of probability density functions (pdfs) belonging to the class of dispersion models introduced by Jørgensen (1997b). Dispersion models form a broad family that includes many familiar distributions—normal, Student‑t, gamma, inverse Gaussian, hyperbola, von‑Mises, among others—by expressing each as an exponential family with a dispersion (scale) parameter ϕ. The authors focus on the “small‑dispersion” asymptotic regime, i.e., the limit ϕ → 0, where the pdf concentrates around its mean but the shape of the tails becomes critical for extreme‑value analysis, Bayesian posterior approximations, and importance‑sampling design.

A central tool is the uniformly convergent saddlepoint approximation. Starting from the canonical representation

  f(y;θ,ϕ)=exp{