Accurate Inference for the Mean of the Poisson-Exponential Distribution

Authors
Wei Lin 1
Xiang Li 2
Augustine Wong 2
1 Department of Mathematics and Statistics, Thompson Rivers University, 805 TRU Way, Kamloops, British Columbia, Canada V2C 0C8.
2 Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3.
10.29252/jirss.19.1.1
Abstract
Although the random sum distribution has been well-studied in probability theory, inference for the mean of such distribution is very limited in the literature. In this paper, two approaches are proposed to obtain inference for the mean of the Poisson-Exponential distribution. Both proposed approaches require the log-likelihood function of the Poisson-Exponential distribution, but the exact form of the log-likelihood function is not available. An approximate form of the log-likelihood function is then derived by the saddlepoint method. Inference for the mean of the Poisson-Exponential distribution can either be obtained from the modified signed likelihood root statistic or from the Bartlett corrected likelihood ratio statistic. The explicit form of the modified signed likelihood root statistic is derived in this paper, and a systematic method to numerically approximate the Bartlett correction factor, hence the Bartlett corrected likelihood ratio statistic is proposed. Simulation studies show that both methods are extremely accurate even when the sample size is small.
Keywords
Bartlett Correction
p-value Punction
Saddlepoint Approximation
Signed Likelihood Root
  1. Alya, A. M., and Low, H. C. (2013), Saddlepoint approximation to cumulative distribution function for poisson-exponential distribution. Modern Applied Science, 7, 26-32. [DOI:10.5539/mas.v7n3p26]
  2. Barbour, A. D., and Chryssaphinou, O. (2001), Compound poisson approximation: A user's guide. The Annals of Applied Probability, 11, 964-1002. [DOI:10.1214/aoap/1015345355]
  3. Barbour, A. D., Johnson, O., Kontoyiannis, I., and Madiman, M. (2010), Compound poisson approximation via information functionals. Electron. J. Prob., 15, 1344-1369. [DOI:10.1214/EJP.v15-799]
  4. Barndorff-Nielsen, O. E. (1986), Inference on full and partial parameters based on the standardized signed log likelihood ratio. Biometrika, 73, 307-322. [DOI:10.2307/2336207]
  5. Barndorff-Nielsen, O. E. (1991), Modified signed log likelihood ratio. Biometrika, 78, 557-564. [DOI:10.1093/biomet/78.3.557]
  6. Bartlett, M. S. (1937), Properties of sufficiency and statistical tests. Proceedings of the Royal Society of London, 160, 268-282. [DOI:10.1098/rspa.1937.0109]
  7. Bero, R. (2003), Ckerstan's method for compound poisson approximation. The Annals of Probability, 32,1754-1771. [DOI:10.1214/aop/1068646365]
  8. Daniels, H. E. (1954), Saddlepoint approximations in statistics. Annals of Mathematical Statistics, 25, 631-650. [DOI:10.1214/aoms/1177728652]
  9. Daniels, H. E. (1987), Tail probability approximations. International Statistical Review, 55, 37-48. [DOI:10.2307/1403269]
  10. Fraser, D. A. S. (2017), p-values: The insight to modern statistical inference. Annual Review of Statistics and Its Application, 4, 1-14. [DOI:10.1146/annurev-statistics-060116-054139]
  11. Fraser, D. A. S., and Reid, N. (1995), Ancillaries and third-order significance. Utilitas Mathematica, 7, 33-53.
  12. Fraser, D. A. S., Reid, N., and Wu, J. (1999), A simple general formula for tail probabilities for frequentist and bayesian inference. Biometrika, 86, 249-264. [DOI:10.1093/biomet/86.2.249]
  13. Ghribi, A. and Masmoudi, A. (2013), Acompound poisson model for learning discrete bayesian networks. Acta Mathematica Scientia, 33, 1767-1784. [DOI:10.1016/S0252-9602(13)60122-8]
  14. Kalbfleisch, J. G. (1985), Probability and Statistical Inference Volumne 2: Statistical Inference (2nd Edition). Springer-Verlag, New York. [DOI:10.1007/978-1-4612-5136-1]
  15. Lugannani, R., and Rice, S. (1980), Saddlepoint approximation for the distribution of sums of random variables. Advances in Applied Probability, 12, 475-490. [DOI:10.2307/1426607]
  16. Murphy, S. A., and Van der Vaart, A. M. (2000), On profile likelihood. Journal of the American Statistical Association, 95, 449-465. [DOI:10.1080/01621459.2000.10474219]
  17. Reid, N. (2010), Likelihood inference. Wiley Interdisciplinary Reviews: Computational Statistics, 2(5), 517-525. [DOI:10.1002/wics.110]
  18. Thmazella , V. L. D., Cancho, V. G., and Louzada V. (2013), Bayesian reference analysis for the poisson-exponential lifetime distribution. Chilean Journal of Statistics, 4, 99-113.
Journal of the Iranian Statistical Society
Volume 19, Issue 1
June 2020
Pages 1-19

  • Receive Date 23 July 2022