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New Expressions for Repeated Upper Tail Integrals of the Normal Distribution

Author

Listed:
  • Christopher S. Withers

    (Industrial Research Limited)

  • Saralees Nadarajah

    (University of Manchester)

Abstract

Expressions are given for repeated upper tail integrals of the univariate normal density (and so also for the general Hermite function) for both positive and negative arguments. The expressions involve moments of the form E(x + i N) n and E1 / (x 2 + N 2) n , where N is a unit normal random variable. Laplace transforms are provided for the Hermite functions and the moments. The practical use of these expressions is illustrated.

Suggested Citation

  • Christopher S. Withers & Saralees Nadarajah, 2011. "New Expressions for Repeated Upper Tail Integrals of the Normal Distribution," Methodology and Computing in Applied Probability, Springer, vol. 13(4), pages 855-871, December.
    • Handle: RePEc:spr:metcap:v:13:y:2011:i:4:d:10.1007_s11009-010-9198-3
    DOI: 10.1007/s11009-010-9198-3
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    References listed on IDEAS

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    1. Presnell, Brett & Rumcheva, Pavlina, 2008. "The mean resultant length of the spherically projected normal distribution," Statistics & Probability Letters, Elsevier, vol. 78(5), pages 557-563, April.
    2. Withers, C.S. & McGavin, P.N., 2006. "Expressions for the normal distribution and repeated normal integrals," Statistics & Probability Letters, Elsevier, vol. 76(5), pages 479-487, March.
    3. Withers, C. S., 2000. "A simple expression for the multivariate Hermite polynomials," Statistics & Probability Letters, Elsevier, vol. 47(2), pages 165-169, April.
    4. K. V. Mardia, 1998. "Fisher's repeated normal integral function and shape distributions," Journal of Applied Statistics, Taylor & Francis Journals, vol. 25(2), pages 231-235.
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