A Stein characterisation of the generalized hyperbolic distribution

Gaunt, Robert E.

ESAIM: Probability and Statistics, Tome 21 (2017), pp. 303-316.

The generalized hyperbolic (GH) distributions form a five parameter family of probability distributions that includes many standard distributions as special or limiting cases, such as the generalized inverse Gaussian distribution, Student’s $t$-distribution and the variance-gamma distribution, and thus the normal, gamma and Laplace distributions. In this paper, we consider the GH distribution in the context of Stein’s method. In particular, we obtain a Stein characterisation of the GH distribution that leads to a Stein equation for the GH distribution. This Stein equation reduces to the Stein equations from the current literature for the aforementioned distributions that arise as limiting cases of the GH superclass.

B. Arras, E. Azmoodeh, G. Poly and Y. Swan, Stein’s method on the second Wiener chaos: 2-Wasserstein distance. Preprint (2016). | arXiv

A.D. Barbour, Stein’s method for diffusion approximations. Probab. Theory Rel. 84 (1990) 297–322. | DOI | MR | Zbl

A.D. Barbour, L. Holst and S. Janson, Poisson Approximation. Oxford University Press, Oxford (1992). | MR | Zbl

O.E. Barndorff-Nielsen, Exponentially decreasing distributions for the logarithm of particle size. Proc. R. Soc. London A 353 (1977) 401–419. | DOI

O.E. Barndorff-Nielsen, J. Kent and M. Sørensen, Normal Variance-Mean Mixtures and $z$-Distributions. Int. Stat. Rev. 50 (1982) 145–159. | DOI | MR | Zbl

B.M. Bibby and M. Sørensen, Hyperbolic Processes in Finance. In Handbook of Heavy Tailed Distributions in Finance. Edited by S. Rachev. Elsevier Science, Amsterdam (2003) 211–248.

S. Chatterjee, J. Fulman and A. Röllin, Exponential approximation by Stein’s method and spectral graph theory. ALEA Lat. Am. J. Probab. Math. Stat. 8 (2011) 197–223. | MR | Zbl

S. Chatterjee and Q.–M. Shao, Nonnormal approximation by Steins method of exchangeable pairs with application to the Curie-Weiss model. Ann. Appl. Probab. 21 (2011) 464–483. | DOI | MR | Zbl

L.H.Y. Chen, Poisson approximation for dependent trials. Ann. Probab. 3 (1975) 534–545. | MR | Zbl

L.H.Y. Chen, L. Goldstein and Q-M. Shao, Normal Approximation by Stein’s Method. Springer (2011). | MR | Zbl

P.J. Collins, Differential and Integral Equations. Oxford University Press (2006). | Zbl

P. Diaconis and S. Zabell, Closed Form Summation for Classical Distributions: Variations on a Theme of De Moivre. Statist. Sci. 6 (1991) 284–302. | DOI | MR | Zbl

C. Döbler, Stein’s method of exchangeable pairs for the beta distribution and generalizations. Electron. J. Probab. 20 (2015) 1–34. | DOI | MR | Zbl

C. Döbler, Distributional transformations without orthogonality relations. J. Theor. Probab. 30 (2017) 85–116. | DOI | MR | Zbl

C.Döbler, R.E. Gaunt and S.J. Vollmer, An iterative technique for bounding derivatives of solutions of Stein equations. Preprint (2015). | arXiv | MR

E. Eberlein and E. Hammerstein, Generalized Hyperbolic and Inverse Gaussian Distributions: Limiting Cases and Approximation of Processes. In Seminar on Stochastic Analysis, Random Fields and Applications IV, edited by R.C.Dalang, M. Dozzi, F. Russo. In vol. 58 of Progress in Probability. Birkhäuser Verlag (2004) 105–153. | MR

E. Eberlein and U. Keller, Hyperbolic distributions in finance. Bernoulli 1 (1995) 281–299. | DOI | Zbl

E. Eberlein and K. Prause, The generalized hyperbolic model: financial derivatives and risk measures. In Mathematical finance – Bachelier Congress 2000. Edited by H. Geman, D.B. Madan, S. Pliska and T. Vorst. Springer, Berlin (2001) 245–267. | MR | Zbl

P. Eichelsbacher and B. Martschink, Rates of Convergence in the Blume-Emery-Griffiths Model. J. Stat. Phys. 154 (2014) 1483–1507. | DOI | MR | Zbl

P. Eichelsbacher and C. Thäle, Malliavin-Stein method for Variance-Gamma approximation on Wiener space. Electron. J. Probab. 20 (2015) 1–28. | DOI | MR | Zbl

R.E. Gaunt, Variance-Gamma approximation via Stein’s method. Electron. J. Probab. 19 1–33. | MR | Zbl

Gaunt, R.E. Stein’s method for functions of multivariate normal random variables. Preprint (2015). | arXiv | MR

R.E. Gaunt, Uniform bounds for expressions involving modified Bessel functions. Math. Inequal. Appl. 19 (2016) 1003–1012. | MR | Zbl

R.E. Gaunt, On Stein’s method for products of normal random variables and zero bias couplings. Bernouilli 23 (2017) 3311–3345. | DOI | MR | Zbl

R.E. Gaunt, Products of normal, beta and gamma random variables: Stein operators and distributional theory. To appear in Braz. J. Probab. Stat. (2016). | MR

R.E. Gaunt, A. Pickett and G. Reinert, Chi-square approximation by Stein’s method with application to Pearson’s statistic. To appear in Ann. Appl. Probab. (2017). | MR

R.E. Gaunt and G. Reinert, The rate of convergence of some asymptotically chi-square distributed statistics by Stein’s method. Preprint (2016). | arXiv

L. Goldstein and G. Reinert, Stein’s method for the Beta distribution and the Pólya-Eggenberger Urn. J. Appl. Probab. 50 (2013) 1187–1205. | DOI | MR | Zbl

F. Götze, On the rate of convergence in the multivariate CLT. Ann. Probab. 19 (1991) 724–739. | DOI | MR | Zbl

A.E. Koudou and C. Ley, Characterizations of GIG laws: a survey complemented with two new results. Probab. Surv. 11 (2014) 161–176. | DOI | MR | Zbl

C. Ley, G. Reinert and Y. Swan, Stein’s method for comparison of univariate distributions. Probab. Surv. 14 (2017) 1–52. | MR | Zbl

H. Luk, Stein’s Method for the Gamma Distribution and Related Statistical Applications. Ph.D. thesis, University of Southern California (1994). | MR

L. Mackey, M.I. Jordan, R.Y. Chen, B. Farrell and J.A. Tropp, Matrix concentration inequalities via the method of exchangeable pairs. Ann. Probab. 42 (2014) 906–945. | DOI | MR | Zbl

D.B. Madan and E. Seneta, The Variance Gamma (V.G.) Model for Share Market Returns. J. Bus. 63 (1990) 511–524. | DOI

F.W.J. Olver, D.W. Lozier, R.F. Boisvert and C.W. Clark, NIST Handbook of Mathematical Functions. Cambridge University Press (2010). | MR | Zbl

E. Peköz and A. Röllin, New rates for exponential approximation and the theorems of Rényi and Yaglom. Ann. Probab. 39 (2011) 587–608. | DOI | MR | Zbl

E. Peköz, A. Röllin and N. Ross, Degree asymptotics with rates for preferential attachment random graphs. Ann. Appl. Probab. 23 (2013) 1188–1218. | DOI | MR | Zbl

E. Peköz, A. Röllin and N. Ross, Generalized gamma approximation with rates for urns, walks and trees. Ann. Probab. 44 (2016) 1776–1816. | DOI | MR | Zbl

J. Pike and H. Ren, Stein’s method and the Laplace distribution. ALEA Lat. Am. J. Probab. Math. Stat. 11 (2014) 571–587. | MR | Zbl

N. Ross, Fundamentals of Stein’s method. Probab. Surv. 8 (2011) 210–293. | DOI | MR | Zbl

W. Schoutens, Orthogonal Polynomials in Steins Method. EURANDOM Report 99-041, Eurandom (1999).

W. Schoutens, Orthogonal polynomials in Stein’s method. J. Math. Anal. Appl. 253 (2001) 515–531. | DOI | MR | Zbl

D.J. Scott, D. Würtz, C. Dong and T.T. Tran, Moments of the generalized hyperbolic distribution. Comput. Stat. 26 (2011) 459–476. | DOI | MR | Zbl

C. Stein, A bound for the error in the normal approximation to the the distribution of a sum of dependent random variables. In In Vol. 2 of Proc. Sixth Berkeley Symp. Math. Statis. Prob. Univ. California Press, Berkeley (1972) 583–602. | MR | Zbl

C. Stein, Approximate Computation of Expectations. IMS, Hayward, California (1986). | MR | Zbl