New error bounds for Laplace approximation via Stein’s method
ESAIM: Probability and Statistics, Tome 25 (2021), pp. 325-345

We use Stein’s method to obtain explicit bounds on the rate of convergence for the Laplace approximation of two different sums of independent random variables; one being a random sum of mean zero random variables and the other being a deterministic sum of mean zero random variables in which the normalisation sequence is random. We make technical advances to the framework of Pike and Ren [ALEA Lat. Am. J. Probab. Math. Stat. 11 (2014) 571–587] for Stein’s method for Laplace approximation, which allows us to give bounds in the Kolmogorov and Wasserstein metrics. Under the additional assumption of vanishing third moments, we obtain faster convergence rates in smooth test function metrics. As part of the derivation of our bounds for the Laplace approximation for the deterministic sum, we obtain new bounds for the solution, and its first two derivatives, of the Rayleigh Stein equation.

DOI : 10.1051/ps/2021012
Classification : 60F05, 62E17
Keywords: Stein’s method, Laplace approximation, rate of convergence, random sums, Rayleigh distribution 
@article{PS_2021__25_1_325_0,
     author = {Gaunt, Robert E.},
     title = {New error bounds for {Laplace} approximation \protect\emph{via} {Stein{\textquoteright}s} method},
     journal = {ESAIM: Probability and Statistics},
     pages = {325--345},
     year = {2021},
     publisher = {EDP-Sciences},
     volume = {25},
     doi = {10.1051/ps/2021012},
     mrnumber = {4291370},
     zbl = {1482.60036},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/ps/2021012/}
}
TY  - JOUR
AU  - Gaunt, Robert E.
TI  - New error bounds for Laplace approximation via Stein’s method
JO  - ESAIM: Probability and Statistics
PY  - 2021
SP  - 325
EP  - 345
VL  - 25
PB  - EDP-Sciences
UR  - https://www.numdam.org/articles/10.1051/ps/2021012/
DO  - 10.1051/ps/2021012
LA  - en
ID  - PS_2021__25_1_325_0
ER  - 
%0 Journal Article
%A Gaunt, Robert E.
%T New error bounds for Laplace approximation via Stein’s method
%J ESAIM: Probability and Statistics
%D 2021
%P 325-345
%V 25
%I EDP-Sciences
%U https://www.numdam.org/articles/10.1051/ps/2021012/
%R 10.1051/ps/2021012
%G en
%F PS_2021__25_1_325_0
Gaunt, Robert E. New error bounds for Laplace approximation via Stein’s method. ESAIM: Probability and Statistics, Tome 25 (2021), pp. 325-345. doi: 10.1051/ps/2021012

[1] B. Arras and C. Houdré, On Stein’s method for infinitely divisible laws with finite first moment. Springer Briefs in Probability and Mathematical Statistics. Springer, Cham (2019). | MR

[2] A.C. Berry, The accuracy of the Gaussian approximation to the sum of independent variates. Trans. Am. Math. Soc. 49 (1941) 122–136. | MR | JFM | Zbl | DOI

[3] A. Braverman and J.G. Dai, High order steady-state diffusion approximation of the Erlang-C system. Preprint (2016). | arXiv

[4] L.H.Y. Chen and Q.-M. Shao, Stein’s method for normal approximation, in Vol. 4 of Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. Singapore Univ. Press, Singapore (2005) 1–59. | MR

[5] F. Daly, Upper bounds for Stein-type operators. Electon. J. Probab. 13 (2008) 566–587. | MR | Zbl

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

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

[8] C. Döbler, R.E. Gaunt and S.J. Vollmer, An iterative technique for bounding derivatives of solutions of Stein equations. Electr. J. Probab. 22 (2017) 1–39. | MR | Zbl

[9] L. Döbler, R. Samworth and J. Wellner, Bounding distributional errors via density ratios. Preprint (2019). | arXiv | MR | Zbl

[10] M. Ernst, G. Reinert and Y. Swan, First order covariance inequalities via Stein’s method. To appear in Bernoulli (2020). | MR | Zbl

[11] M. Ernst and Y. Swan, Distances between distributions via Stein’s method. Preprint (2019). | arXiv | MR | Zbl

[12] C.G. Esseen, On the Liapunoff limit of error in the theory of probability. Ark. Mat. Astron. Fys. A28 (1942) 1–19. | MR | JFM | Zbl

[13] M. Fathi, Higher-order Stein kernels for Gaussian approximation. To appear in Stud. Math. (2020). | MR | Zbl

[14] W. Feller, On the Berry-Esseen Theorem. Z. Wahrscheinlichkeit 10 (1968) 261–268. | MR | Zbl | DOI

[15] R.E. Gaunt, Variance-Gamma approximation via Stein’s method. Electr. J. Probab. 19 (2014) 1–33. | MR | Zbl

[16] R.E. Gaunt, Rates of convergence in normal approximation under moment conditions via new bounds on solutions of the Stein equation. J. Theor. Probab. 29 (2016) 231–247. | MR | Zbl | DOI

[17] R.E. Gaunt, Products of normal, beta and gamma random variables: Stein operators and distributional theory. Braz. J. Probab. Stat. 32 (2018) 437–466. | MR | Zbl | DOI

[18] R.E. Gaunt Wasserstein and Kolmogorov error bounds for variance-gamma approximation via Stein’s method I. J. Theor. Probab. 33 (2020) 465–505. | MR | Zbl | DOI

[19] R.E. Gaunt Stein’s method for functions of multivariate normal random variables. Ann. Inst. Henri Poincaré Prob. Stat. 56 (2020) 1484–1513. | MR | Zbl | DOI

[20] R.E. Gaunt, G. Mijoule, and Y. Swan, An algebra of Stein operators. J. Math. Anal. Appl. 469 (2019) 260–279. | MR | Zbl | DOI

[21] 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. | MR | Zbl | DOI

[22] 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 | DOI

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

[24] M.E.H. Ismail, L. Lorch and M.E. Muldoon, Completely monotonic functions associated with the gamma function and its q -analogues. J. Math. Anal. Appl. 116 (1986) 1–9. | MR | Zbl | DOI

[25] A.V. Kakosyan, L.B. Klebanov and I.A. Melamed, Characterization of Distributions by the Method of Intensively Monotone Operators, Vol. 1088 of Lecture Notes in Math. Springer, Berlin (1984). | MR | Zbl

[26] V. Kalashnikov, Geometric Sums: Bounds for Rare Events with Applications. Risk Analysis, Reliability, Queueing. Kluwer Academic Publishers Group, Dordrecht (1997). | MR | Zbl

[27] E. Konzou and A. Koudou, About the Stein equation for the generalized inverse Gaussian and Kummer distributions. ESAIM: PS 24 (2020) 607–626. | MR | Zbl | Numdam | DOI

[28] S. Kotz, T.J. Kozubowski and K. Podgórski, The Laplace Distribution and Generalizations: A Revisit with New Applications. Springer (2001). | MR | Zbl | DOI

[29] C. Lefèvre and S. Utev, Exact norms of a Stein-type operator and associated stochastic orderings. Probab. Theory Rel. 127 (2003) 353–366. | MR | Zbl | DOI

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

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

[32] 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

[33] A.G. Pakes, A characterization of gamma mixtures of stable laws motivated by limit theorems. Stat. Neerl. 46 (1992) 209–218. | MR | Zbl | DOI

[34] A.G. Pakes, On characterizations through mixed sums. Aust. J. Stat. 34 (1992) 323–339. | MR | Zbl | DOI

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

[36] E. Peköz, A. Röllin and N. Ross, Total variation error bounds for geometric approximation. Bernoulli 19 (2013) 610–632. | MR | Zbl | DOI

[37] E. Peköz, A. Röllin and N. Ross, Exponential and Laplace approximation for occupation statistics of branching random walk. Electr. J. Probab. 25 (2020) 1–22. | MR | Zbl

[38] 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

[39] G. Reinert, Couplings for normal approximations with Stein’s method, in Microsurveys in Discrete Probability, volume of DIMACS series AMS (1998) 193–207. | MR | Zbl

[40] A. Rényi, A characterization of Poisson processes. Magyar Tud. Akad. Mat. Kutató Int. Közl. 1 (1957) 519–527. | MR | Zbl

[41] W. Schoutens, Orthogonal Polynomials in Steins Method. EURANDOM Report 99-041, EURANDOM, 1999. | MR | Zbl

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

[43] I. Shevtsova, An improvement of convergence rate estimates in the Lyapunov theorem. Dokl. Math. 253 (2010) 862–864. | Zbl | DOI

[44] I. Shevtsova, On the absolute constants in the Berry Esseen type inequalities for identically distributed summands. Preprint (2011). | arXiv

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

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

[47] A.A. Toda, Weak limit of the geometric sum of independent but not identically distributed random variables. Preprint (2011). | arXiv

Cité par Sources :