Zero bias transformation and asymptotic expansions
Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 1, pp. 258-281.

Soit W une somme de variables aléatoires indépendants. On applique la transformation zéro biais pour obtenir de façon recursive des développements asymptotiques de 𝔼[h(W)] en terme d’espérances par rapport à la loi normale, ou à la loi de Poisson si les variables aléatoires sont à valeurs entières. On discute aussi les bornes des termes d’erreur.

Let W be a sum of independent random variables. We apply the zero bias transformation to deduce recursive asymptotic expansions for 𝔼[h(W)] in terms of normal expectations, or of Poisson expectations for integer-valued random variables. We also discuss the estimates of remaining errors.

DOI : 10.1214/10-AIHP384
Classification : 60G50, 60F05
Mots-clés : normal and Poisson approximations, zero bias transformation, Stein's method, reverse Taylor formula, concentration inequality 
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Jiao, Ying. Zero bias transformation and asymptotic expansions. Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 1, pp. 258-281. doi : 10.1214/10-AIHP384. http://www.numdam.org/articles/10.1214/10-AIHP384/

[1] R. Arratia, L. Goldstein and L. Gordon. Two moments suffice for Poisson approximations: The Chen-Stein method. Ann. Probab. 17 (1989) 9-25. | MR | Zbl

[2] A. D. Barbour. Asymptotic expansions based on smooth functions in the central limit theorem. Probab. Theory Related Fields 72 (1986) 289-303. | MR | Zbl

[3] A. D. Barbour. Asymptotic expansions in the Poisson limit theorem. Ann. Probab. 15 (1987) 748-766. | MR | Zbl

[4] A. D. Barbour and V. Čekanavičius. Total variation asymptotics for sums of independent integer random variables. Ann. Probab. 30 (2002) 509-545. | MR | Zbl

[5] A. D. Barbour, L. H. Y. Chen and K. P. Choi. Poisson approximation for unbounded functions. I. Independent summands. Statist. Sinica 5 (1995) 749-766. | MR | Zbl

[6] A. D. Barbour, L. Holst and S. Janson. Poisson Approximation. Oxford Univ. Press, Oxford, 1992. | MR | Zbl

[7] L. H. Y. Chen. Poisson approximation for dependent trials. Ann. Probab. 3 (1975) 534-545. | MR | Zbl

[8] L. H. Y. Chen and Q.-M. Shao. A non-uniform Berry-Esseen bound via Stein's method. Probab. Theory Related Fields 120 (2001) 236-254. | MR | Zbl

[9] L. H. Y. Chen and Q.-M. Shao. Stein's method for normal approximation. In An Introduction to Stein's Method 1-59. Lecture Notes Series, IMS, National University of Singapore 4. Singapore Univ. Press, Singapore, 2005. | MR

[10] L. H. Y. Chen and Q.-M. Shao. Normal approximation for nonlinear statistics using a concentration inequality approach. Bernoulli 13 (2007) 581-599. | MR | Zbl

[11] N. El Karoui and Y. Jiao. Stein's method and zero bias transformation for CDOs tranches pricing. Finance Stoch. 13 (2009) 151-180. | MR | Zbl

[12] T. Erhardsson. Stein's method for Poisson and compound Poisson approximation. In An Introduction to Stein's Method 61-113. Lecture Notes Series, IMS, National University of Singapore 4. Singapore Univ. Press, Singapore, 2005. | MR

[13] L. Goldstein. L1 bounds in normal approximation. Ann. Probab. 35 (2007) 1888-1930. | MR | Zbl

[14] L. Goldstein. Bounds on the constant in the mean central limit theorem. Ann. Probab. 38 (2010) 1672-1689. | MR | Zbl

[15] L. Goldstein and G. Reinert. Stein's method and the zero bias transformation with application to simple random sampling. Ann. Appl. Probab. 7 (1997) 935-952. | MR | Zbl

[16] L. Goldstein and G. Reinert. Distributional transformations, orthogonal polynomials, and stein characterizations. J. Theoret. Probab. 18 (2005) 237-260. | MR | Zbl

[17] F. Götze and C. Hipp. Asymptotic expansions in the central limit theorem under moment conditions. Z. Wahrsch. Verw. Gebiete 42 (1978) 67-87. | MR | Zbl

[18] C. Hipp. Edgeworth expansions for integrals of smooth functions. Ann. Probab. 5 (1977) 1004-1011. | MR | Zbl

[19] Y. Jiao. Risque de crédit: modélisation et simulation numérique. PhD thesis, Ecole Polytechnique, 2006. Available at http://www.imprimerie.polytechnique.fr/Theses/Files/Ying.pdf. | Zbl

[20] A. Kolmogolov and S. Fomine. Éléments de la théorie des fonctions et de l'analyse fonctionnelle. Éditions Mir., Moscow, 1974. | MR | Zbl

[21] V. V. Petrov. Sums of Independent Random Variables. Springer, New York, 1975. | MR | Zbl

[22] V. Rotar. Stein's method, Edgeworth's expansions and a formula of Barbour. In Stein's Method and Applications 59-84. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 5. Singapore Univ. Press, Singapore, 2005. | MR

[23] P. Smith. A recursive formulation of the old problem of obtaining moments from cumulants and vice versa. Amer. Statist. 49 (1995) 217-218. | MR

[24] C. Stein. A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proc. Sixth Berkeley Symp. Math. Statist. Probab. 583-602. California Univ. Press, Berkeley, 1972. | MR | Zbl

[25] C. Stein. Approximate Computation of Expectations. IMS, Hayward, CA, 1986. | MR | Zbl

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