About the Stein equation for the generalized inverse Gaussian and Kummer distributions
ESAIM: Probability and Statistics, Tome 24 (2020), pp. 607-626.

We observe that the density of the Kummer distribution satisfies a certain differential equation, leading to a Stein characterization of this distribution and to a solution of the related Stein equation. A bound is derived for the solution and for its first and second derivatives. To provide a bound for the solution we partly use the same framework as in Gaunt 2017 [Stein, ESAIM: PS 21 (2017) 303–316] in the case of the generalized inverse Gaussian distribution, which we revisit by correcting a minor error. We also bound the first and second derivatives of the Stein equation in the latter case.

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DOI : 10.1051/ps/2020009
Classification : 60F05, 60E05
Mots-clés : Generalized inverse Gaussian distribution, Kummer distribution, Stein characterization 
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     title = {About the {Stein} equation for the generalized inverse {Gaussian} and {Kummer} distributions},
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Konzou, Essomanda; Koudou, Angelo Efoevi. About the Stein equation for the generalized inverse Gaussian and Kummer distributions. ESAIM: Probability and Statistics, Tome 24 (2020), pp. 607-626. doi : 10.1051/ps/2020009. http://www.numdam.org/articles/10.1051/ps/2020009/

[1] L.H.Y. Chen, L. Goldstein and Q.-M. Shao, Normal approximation by Stein’s method. Probability and its Applications Heidelberg (2011). | MR | Zbl

[2] R.E. Gaunt, A Stein characterization of the generalized hyperbolic distribution. ESAIM: PS 21 (2017) 303–316. | DOI | Numdam | MR | Zbl

[3] R.E. Gaunt, A.M. Pickett and G. Reinert, Chi-square approximation by Stein’s method with application to Pearson’s statistic. Ann. Appl. Probab. 27 (2017) 720–756. | DOI | MR | Zbl

[4] L. Goldstein and G. Reinert, Stein’s method for the Beta distribution and the Pòlya-Eggenberger urn. Adv. Appl. Probab. 50 (2013) 1187–1205. | DOI | MR | Zbl

[5] M. Hamza and P. Vallois, On Kummer’s distribution of type two and a generalized beta distribution. Statist. Prob. Lett. 118 (2016) 60–69. | DOI | MR | Zbl

[6] A.E. Koudou and P. Vallois, Independence properties of the Matsumoto-Yor type. Bernoulli 18 (2012) 119–136. | DOI | MR | Zbl

[7] 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

[8] G. Letac and V. Seshadri, A characterization of the generalized inverse Gaussian distribution by continued fractions. Z. Wahr. verw. Geb. 62 (1983) 485–489. | DOI | MR | Zbl

[9] C. Ley and Y. Swan, Stein’s density approach and information inequalities. Electron. Comm. Probab. 18 (2013) 1–14. | MR | Zbl

[10] A. Piliszek and J. Wesołowski, Change of measure technique in characterizations of the gamma and Kummer distributions. J. Math. Anal. Appl. 458 (2018) 967–979. | DOI | MR | Zbl

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

[12] W. Schoutens, Orthogonal polynomials in steins method. J. Math. Anal. Appl. 253 (2001) 515–531. | DOI | MR | Zbl

[13] Q.-M. Shao and Z.-S. Zhang, Identifying the limiting distribution by a general approach of Stein’s method. Sci. China Math. 59 (2016) 2379–2392. | DOI | MR | Zbl

[14] C. Stein, A bound for the error in the normal approximation to the distribution of a sum of dependent random variables, in Vol. 2 of Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability. University of California Press, Berkeley (1972) 583–602. | MR | Zbl

[15] C. Stein, P. Diaconis, S. Holmes and G. Reinert, Use of exchangeable pairs in the analysis of simulations, in Stein’s method: expository lectures and applications, edited by Persi Diaconis and Susan Holmes. Vol. 46 of IMS Lecture Notes Monogr. Ser. Institute of Mathematical Statistics Beachwood, Ohio, USA (2004) 1–26. | MR

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