Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

Krawtchouk Polynomial Approximation for Binomial Convolutions

  • Ha, Hyung-Tae (Department of Applied Statistics, Gachon University)
  • Received : 2016.07.07
  • Accepted : 2017.08.09
  • Published : 2017.10.23
Download PDF
⟨ Previous Next ⟩

Abstract

We propose an accurate approximation method via discrete Krawtchouk orthogonal polynomials to the distribution of a sum of independent but non-identically distributed binomial random variables. This approximation is a weighted binomial distribution with no need for continuity correction unlike commonly used density approximation methods such as saddlepoint, Gram-Charlier A type(GC), and Gaussian approximation methods. The accuracy obtained from the proposed approximation is compared with saddlepoint approximations applied by Eisinga et al. [4], which are the most accurate method among higher order asymptotic approximation methods. The numerical results show that the proposed approximation in general provide more accurate estimates over the entire range for the target probability mass function including the right-tail probabilities. In addition, the method is mathematically tractable and computationally easy to program.

Keywords

References

  1. J. C. Benneyan and A. TaSeli, Exact and approximate probability distributions of evidence-based bundle composite compliance measures, Health Care Manag. Sci., 13(2010), 193-209. https://doi.org/10.1007/s10729-009-9123-x
  2. K. Butler and M. A. Stephens, The distribution of a sum of independent binomial random variables, Methodol Comput. Appl. Probab., in press, doi:10.1007/s11009-016-9533-4.
  3. R. W. Butler, Saddlepoint Approximations with Applications, Cambridge University Press, Cambridge, 2007.
  4. R. Eisinga, M. T. Grotenhuis and B. Pelzer, Saddlepoint approximations for the sum of independent non-identically distributed binomial random variables, Stat. Neerl., 67(2013), 190-201. https://doi.org/10.1111/stan.12002
  5. M. B. Gordy, Saddlepoint approximation of $CreditRisk^+$, J. Bank. Financ., 26(2002), 1335-1353. https://doi.org/10.1016/S0378-4266(02)00266-2
  6. Y. Hong, On computing the distribution function for the sum of independent and non-identical random indicators, Technical Report No. 11-2, Department of Statistics, Virginia Tech, Blacksburg, VA, 2011.
  7. S. H. Ong, Some stochastic models leading to the convolution of two binomial variables, Stat. Probab. Lett., 22(1995), 161-166. https://doi.org/10.1016/0167-7152(94)00063-E
  8. B. K. Shah, On the distribution of the sum of independent integer valued random variables, Am. Stat., 27(1973), 123-124. https://doi.org/10.1080/00031305.1973.10479011
  9. S. L. Smalley, J. A. Woodward and C. G. S. Palmer, A general statistical model for detecting complex-trait loci by using affected relative pairs in a genome search, Am. J. Hum. Genet., 58(1996), 844-860.