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New Expressions for Repeated Upper Tail Integrals of the Normal Distribution

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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.

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References

  • Abramowitz M, Stegun IA (1964) Handbook of mathematical functions. Applied mathematics series, vol 55. U.S. Department of Commerce, National Bureau of Standards

  • Comtet L (1974) Advanced combinatorics. Reidel, Dordrecht

    MATH  Google Scholar 

  • Fisher RA (1931) Introduction of “Table of Hh functions”, of Airey (1931), xxvi–xxxvii. Mathematical tables, vol 1, 2nd edn. (1946), 3rd edn. (1951). British Association for the Advancement of Science, London

  • Goodall CR, Mardia KV (1991) A geometric derivation of the shape density. Adv Appl Probab 23:496–514

    Article  MathSciNet  MATH  Google Scholar 

  • Gradshteyn IS, Ryzhik IM (1965) Tables of integrals, series and products, 4th edn. Academic, New York

    Google Scholar 

  • Hansen ER (1975) A table of series and products. Prentice-Hall, Englewood Cliffs

    MATH  Google Scholar 

  • Mardia KV (1998) Fisher’s repeated normal integral function and shape distributions. J Appl Stat 25:231–235

    Article  MathSciNet  MATH  Google Scholar 

  • Presnell B, Rumcheva P (2008) The mean resultant length of the spherically projected normal distribution. Stat Probab Lett 78:557–563

    Article  MathSciNet  MATH  Google Scholar 

  • Withers CS (2000) A simple expression for the multivariate Hermite polynomials. Stat Probab Lett 47:165–169

    Article  MathSciNet  MATH  Google Scholar 

  • Withers CS, McGavin P (2006) Expressions for the normal distribution and repeated normal integrals. Stat Probab Lett 76:479–487

    Article  MathSciNet  MATH  Google Scholar 

  • Withers CS, Nadarajah S (2007) New expressions for repeated lower tail integrals of the normal distribution. J Korean Stat Soc 36:411–421

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Applied Mathematics Group, Industrial Research Limited, Lower Hutt, New Zealand

    Christopher S. Withers

  2. School of Mathematics, University of Manchester, Manchester, M13 9PL, UK

    Saralees Nadarajah

Authors
  1. Christopher S. Withers
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  2. Saralees Nadarajah
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Correspondence to Saralees Nadarajah.

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Withers, C.S., Nadarajah, S. New Expressions for Repeated Upper Tail Integrals of the Normal Distribution. Methodol Comput Appl Probab 13, 855–871 (2011). https://doi.org/10.1007/s11009-010-9198-3

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  • DOI: https://doi.org/10.1007/s11009-010-9198-3

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