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Efficient simulation from a gamma distribution with small shape parameter

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Abstract

Simulating from a gamma distribution with small shape parameter is a challenging problem. Towards an efficient method, we obtain a limiting distribution for a suitably normalized gamma distribution when the shape parameter tends to zero. Then this limiting distribution provides insight to the construction of a new, simple, and highly efficient acceptance–rejection algorithm. The proposed method is fast and comparisons based on acceptance rates show that it is more efficient than existing acceptance–rejection methods.

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References

  • Abramowitz M, Stegun IA (1966) Handbook of mathematical functions, with formulas, graphs, and mathematical tables. Dover Publications Inc., New York

    MATH  Google Scholar 

  • Ahrens JH, Dieter U (1974) Computer methods for sampling from gamma, beta, Poisson and binomial distributions. Computing (Arch. Elektron. Rechnen) 12(3):223–246

    MATH  MathSciNet  Google Scholar 

  • Best DJ (1983) A note on gamma variate generators with shape parameter less than unity. Computing 30(2):185–188

    Article  MATH  MathSciNet  Google Scholar 

  • Billingsley P (1995) Probability and measure, 3rd edn. Wiley, New York

    MATH  Google Scholar 

  • Brown LD (1986) Fundamentals of statistical exponential families with applications in statistical decision theory. In: Institute of mathematical statistics. Lecture notes—monograph Series, vol 9. Institute of Mathematical Statistics, Hayward

  • Devroye L (1986) Nonuniform random variate generation. Springer, New York

    Book  MATH  Google Scholar 

  • Flury BD (1990) Acceptance-rejection sampling made easy. SIAM Rev 32(3):474–476

    Article  MATH  MathSciNet  Google Scholar 

  • Fraser DAS, Reid N, Wong A (1997) Simple and accurate inference for the mean of a gamma model. Can J Stat 25(1):91–99

    Article  MATH  Google Scholar 

  • Gelman A (2006) Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper). Bayesian Anal 1(3):515–533

    Article  MATH  MathSciNet  Google Scholar 

  • Givens GH, Hoeting JA (2005) Computational statistics. In: Wiley series in probability and statistics. Wiley, Hoboken

  • Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions, vol 1, 2nd edn. In: Wiley series in probability and mathematical statistics: applied probability and statistics. Wiley, New York

  • Kleiber C, Kotz S (2003) Statistical size distributions in economics and actuarial sciences. In: Wiley series in probability and statistics. Wiley, Hoboken

  • Kundu D, Gupta RD (2007) A convenient way of generating gamma random variables using generalized exponential distribution. Comput Stat Data Anal 51(6):2796–2802

    Article  MATH  MathSciNet  Google Scholar 

  • Lange K (1999) Numerical analysis for statisticians. Springer, New York (Statistics and Computing)

    MATH  Google Scholar 

  • Lunn D, Jackson C, Best N, Thomas A, Spiegelhalter D (2012) The BUGS book. CRC Press/Chapman & Hall, Boca Raton

    MATH  Google Scholar 

  • Marin J-M, Pudlo P, Robert CP, Ryder RJ (2012) Approximate Bayesian computational methods. Stat Comput 22(6):1167–1180

    Article  MATH  MathSciNet  Google Scholar 

  • Marsaglia G, Tsang WW (2000) A simple method for generating gamma variables. ACM Trans Math Softw 26(3):363–372

    Article  MATH  MathSciNet  Google Scholar 

  • Martin R, Liu C (2015) Marginal inferential models: prior-free probabilistic inference on interest parameters. J Am Stat Assoc 110:1621–1631

    Article  MATH  MathSciNet  Google Scholar 

  • R Core Team (2013) R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna

  • Tanizaki H (2008) A simple gamma random number generator for arbitrary shape parameters. Econ Bull 3:1–10

    Google Scholar 

  • Xi B, Tan KM, Liu C (2013) Logarithmic transformation based gamma random number generators. J Stat Softw 55(4):1–17

    Article  Google Scholar 

Download references

Acknowledgments

The authors thank the anonymous reviewers for their helpful comments on a previous version of this manuscript. This work was partially supported by the U.S. National Science Foundation, Grants DMS–1208833 and DMS–1208841.

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Authors and Affiliations

  1. Department of Statistics, Purdue University, West Lafayette, IN, USA

    Chuanhai Liu

  2. Department of Statistics, North Carolina State University, Raleigh, NC, USA

    Ryan Martin & Nick Syring

Authors
  1. Chuanhai Liu
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  2. Ryan Martin
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  3. Nick Syring
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Correspondence to Ryan Martin.

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Liu, C., Martin, R. & Syring, N. Efficient simulation from a gamma distribution with small shape parameter. Comput Stat 32, 1767–1775 (2017). https://doi.org/10.1007/s00180-016-0692-0

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  • DOI: https://doi.org/10.1007/s00180-016-0692-0

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