[KSCI] Korea Science Citation Index Service

NEW EXPRESSIONS FOR REPEATED LOWER TAIL INTEGRALS OF THE NORMAL DISTRIBUTION

Withers, Christopher S. (Applied Mathematics Group, Industrial Research Limited Lower Hutt)
Nadarajah, Saralees (School of Mathematics, University of Manchester)

Publication Information

Journal of the Korean Statistical Society / v.36, no.3, 2007 , pp. 411-421 More about this Journal

Abstract

The recent work by the authors (see, Withers, 1999; Withers and McGavin, 2006; Withers and Nadarajah, 2006) provided new expressions for repeated upper tail integrals of the univariate normal density and so also for the general Hermite function. Here we derive new expressions for repeated lower tail integrals of the same. The calculations involve the use of Moran's L-function and the Airy function. In particular, the Hermite functions are expressed in terms of Moran's L-function and vice versa.

Keywords

Airy functions; Hermite functions; Hermite polynomials; Mills' ratio; normal distribution; repeated integrals.;

Citations & Related Records

Times Cited By Web Of Science : 1 (Related Records In Web of Science)

Reference

1	MORAN, P. A. P. (1983). 'A new expansion for the multivariate normal distribution', The Australian Journal of Statistics, 25, 339-344 DOI
2	WITHERS, C. S. AND MCGAVIN, P. N. (2006). 'Expressions for the normal distribution and repeated normal integrals', Statistics & Probability Letters, 76, 479-487 DOI ScienceOn
3	AIRY, J. R. (1931). Table of Hh Functions, British Association for the Advancement of Science, London
4	GOODALL, C. R. AND MARDIA, K. V. (1991). 'A geometrical derivation ofthe shape density', Advances in Applied Probability, 23, 496-514 DOI ScienceOn
5	MARDIA, K. V. (1998). 'Fisher's repeated normal integral function and shape distributions', Journal of Applied Statistics, 25, 231-235 DOI
6	FISHER, R. A. (1931). 'Introduction of table of Hh functions', In Table of Hh Functions (Airy, J. R., eds.), British Association for the Advancement of Science, London
7	WITHERS, C. S. (2000). 'A simple expression for the multivariate Hermite polynomials', Statistics & Probability Letters, 47, 165-169 DOI ScienceOn
8	KOTZ, S., BALAKRISHNAN, N. AND JOHNSON, N. L. (2000). Continuous Multivariate Distributions, Volume 1: Models and Applications, 2nd ed., John Wiley & Sons, New York
9	MILLER, K. S. (1975). Multivariate Distributions, Robert E. Krieger Publishing Company, Huntington, New York
10	ABRAMOWITZ, M. AND STEGUN, I. A. (1964). Handbook of Mathematical Functions, U.S. Department of Commerce, National Bureau of Standards, Applied Mathematics Series, 55
11	MILLER, K. S. (1974). Complex Stochastic Processes: An Introduction to Theory and Application, Addison-Wesley Publishing Company, London
12	WITHERS, C. S. AND NADARAJAH, S. (2006). 'New expressions for repeated upper tail integrals for the normal distribution' , preprint