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Polynomial Birth–Death Distribution Approximation in the Wasserstein Distance

Author

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  • Aihua Xia

    (The University of Melbourne)

  • Fuxi Zhang

    (The University of Melbourne)

Abstract

The polynomial birth–death distribution (abbreviated, PBD) on ℐ={0,1,2,…} or ℐ={0,1,2,…,m} for some finite m introduced in Brown and Xia (Ann. Probab. 29:1373–1403, 2001) is the equilibrium distribution of the birth–death process with birth rates {α i } and death rates {β i }, where α i ≥0 and β i ≥0 are polynomial functions of i∈ℐ. The family includes Poisson, negative binomial, binomial, and hypergeometric distributions. In this paper, we give probabilistic proofs of various Stein’s factors for the PBD approximation with α i =a and β i =i+bi(i−1) in terms of the Wasserstein distance. The paper complements the work of Brown and Xia (Ann. Probab. 29:1373–1403, 2001) and generalizes the work of Barbour and Xia (Bernoulli 12:943–954, 2006) where Poisson approximation (b=0) in the Wasserstein distance is investigated. As an application, we establish an upper bound for the Wasserstein distance between the PBD and Poisson binomial distribution and show that the PBD approximation to the Poisson binomial distribution is much more precise than the approximation by the Poisson or shifted Poisson distributions.

Suggested Citation

  • Aihua Xia & Fuxi Zhang, 2009. "Polynomial Birth–Death Distribution Approximation in the Wasserstein Distance," Journal of Theoretical Probability, Springer, vol. 22(2), pages 294-310, June.
    • Handle: RePEc:spr:jotpro:v:22:y:2009:i:2:d:10.1007_s10959-008-0207-1
    DOI: 10.1007/s10959-008-0207-1
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    References listed on IDEAS

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    1. Timothy C. Brown & M. J. Phillips, 1999. "Negative Binomial Approximation with Stein's Method," Methodology and Computing in Applied Probability, Springer, vol. 1(4), pages 407-421, December.
    2. Ehm, Werner, 1991. "Binomial approximation to the Poisson binomial distribution," Statistics & Probability Letters, Elsevier, vol. 11(1), pages 7-16, January.
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