Orthogonal polynomials in Stein’s method. (English) Zbl 0984.62009
The paper systematically develops a relationship between the classical families of orthogonal polynomials and Stein’s method as applied to the distributions in the Pearson and Ord families, that was also discussed by P. Diaconis and S. Zabell [Stat. Sci. 6, No. 3, 284-302 (1991)]. Here, the two are related by way of a generator approach to Stein’s method, in conjunction with the birth and death processes associated with the orthogonal polynomials of S. Karlin and J.L. McGregor’s [Trans. Am. Math. Soc. 85, 489-546 (1957; Zbl 0091.13801)] spectral representation. This leads, in particular, to Stein equations for the Student’s \(t\) and beta distributions.
Reviewer: A.D.Barbour (Zürich)
MSC:
| 62E17 | Approximations to statistical distributions (nonasymptotic) |
| 60J85 | Applications of branching processes |
| 42C15 | General harmonic expansions, frames |
References:
| [1] | Anderson, W. J., Continuous-Time Markov Chains—An Applications-Oriented Approach (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0731.60067 |
| [2] | Andrews, G. E.; Askey, R.; Roy, R., Special Functions. Special Functions, Encyclopedia of Mathematics and Its Applications, 71 (1999), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0920.33001 |
| [3] | Barbour, A. D., Equilibrium distributions for Markov population processes, Adv. Appl. Probab., 12, 591-614 (1980) · Zbl 0434.60084 |
| [4] | Barbour, A. D., Stein’s method and Poisson process convergence, J. Appl. Probab. A, 25, 175-184 (1988) · Zbl 0661.60034 |
| [5] | Barbour, A. D., Stein’s method for diffusion approximations, Probab. Theory Related Fields, 84, 297-322 (1990) · Zbl 0665.60008 |
| [6] | Barbour, A. D.; Holst, L.; Janson, S., Poisson Approximation (1992), Clarendon: Clarendon Oxford · Zbl 0746.60002 |
| [7] | Bhattacharya, R. N.; Waymire, E. C., Stochastic Processes with Applications (1990), Wiley: Wiley New York · Zbl 0744.60032 |
| [8] | Chen, L. H.Y., Poisson approximation for dependent trials, Ann. Probab., 22, 1607-1618 (1975) |
| [9] | Chihara, T. S., An Introduction to Orthogonal Polynomials (1978), Gordon & Breach: Gordon & Breach New York/London/Paris · Zbl 0389.33008 |
| [10] | Diaconis, P.; Zabell, S., Closed form summation for classical distributions: Variations on a theme of de Moivre, Statist. Sci., 6, 284-302 (1991) · Zbl 0955.60500 |
| [11] | Hildebrandt, E. H., Systems of polynomials connected with Charlier expansions and the Pearson differential and difference equation, Ann. Math. Statist., 2, 379-439 (1931) · Zbl 0004.34402 |
| [12] | Johnson, N. L.; Kotz, S.; Kemp, A. W., Univariate Discrete Distributions (1992), Wiley: Wiley New York · Zbl 0773.62007 |
| [13] | Karlin, S.; McGregor, J. L., The differential equations of birth-and-death processes, and the Stieltjes moment problem, Trans. Amer. Math. Soc., 85, 489-546 (1957) · Zbl 0091.13801 |
| [14] | Karlin, S.; Taylor, H. M., A Second Course in Stochastic Processes (1981), Academic Press: Academic Press New York · Zbl 0469.60001 |
| [15] | R. Koekoek, and, R. F. Swarttouw, The Askey-Scheme of Hypergeometric Orthogonal Polynomials and Its \(q\); R. Koekoek, and, R. F. Swarttouw, The Askey-Scheme of Hypergeometric Orthogonal Polynomials and Its \(q\) |
| [16] | Ledermann, W.; Reuter, G. E.H., Spectral theory for the differential equations of simple birth and death processes, Philos. Trans. Roy. Soc. London Ser. A, 246, 321-369 (1954) · Zbl 0059.11704 |
| [17] | H. M. Luk, Stein’s Method for the Gamma Distribution and Related Statistical Applications, Ph.D. Thesis, University of Southern California, Los Angeles, 1994.; H. M. Luk, Stein’s Method for the Gamma Distribution and Related Statistical Applications, Ph.D. Thesis, University of Southern California, Los Angeles, 1994. |
| [18] | Nikiforov, A. F.; Suslov, S. K.; Uvarov, V. B., Classical Orthogonal Polynomials of a Discrete Variable (1991), Springer-Verlag: Springer-Verlag Berlin/Heidelberg · Zbl 0743.33001 |
| [19] | Schoutens, W., Stochastic Processes and Orthogonal Polynomials. Stochastic Processes and Orthogonal Polynomials, Lecture Notes in Statistics, 146 (2000), Springer-Verlag: Springer-Verlag New York · Zbl 0960.60076 |
| [20] | Stein, C., A bound for the error in the normal approximation to the distibution of a sum of dependent random variables, Proc. Sixth Berkeley Symp. Math. Statist. Probab (1972), Univ. of California Press: Univ. of California Press Berkeley, p. 583-602 · Zbl 0278.60026 |
| [21] | Stein, C., Approximate Computation of Expectations. Approximate Computation of Expectations, IMS Lecture Notes, Monograph Series, 7 (1986), IMS: IMS Hayward · Zbl 0721.60016 |
| [22] | Van Assche, W.; Parthasarathy, P. R.; Lenin, R. B., Spectral representation of four finite birth and death processes, Math. Sci., 24, 105-112 (1999) · Zbl 0971.60085 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
