Stochastic Programming pp 1-7 | Cite as
Types of Asymptotic Approximations for Normal Probability Integrals
Chapter
Abstract
For many problems in reliability and optimization it is necessary to calculate the probabilities of large deviations of normal random vectors. Using the structure of the normal probability density it is possible to derive simple asymptotic approximations for such integrals. Here three types of such approximations are described: approximations for the logarithm of the probabilities, approximations for the probabilities and asymptotic expansions for them.
Keywords
Asymptotic analysis asymptotic expansions Gaussian distribution Laplace method large deviations normal distribution random vectorsPreview
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References
- [1]H. Birndt and W.-D. Richter. Vergleichende Betrachtungen zur Bestimmung des asymptotischen Verhaltens mehrdimensionaler Laplace-Gauß-Integrale. Zeitschrift für Analysis und ihre Anwendunge., 4(3):269–276, 1985.Google Scholar
- [2]N. Bleistein and R.A. Handelsman. Asymptotic Expansions of Integrals. Dover Publications Inc., New York, 1986.Google Scholar
- [3]K. Breitung. Asymptotic approximations for multinormal integrals. Journal of the Engineering Mechanics Division ASC., 110(3):357–366, 1984.CrossRefGoogle Scholar
- [4]K. Breitung. Asymptotic crossing rates for stationary Gaussian vector processes. Stochastic Processes and Application., 29:195–207, 1988.CrossRefGoogle Scholar
- [5]K. Breitung. The extreme value distribution of non-stationary vector processes. In A. H.-S. Ang, M. Shinozuka, and G.I. Schuëller, editors, Proceed-ings of ICOSSAR ’89 5th InVl Conf on structural safety and reliabilit., volume II, pages 1327–1332. American Society of Civil Engineers, 1990.Google Scholar
- [6]K. Breitung. Probability approximations by log likelihood maximization. Journal of the Engineering Mechanics Division ASC., 117(3):457–477, 1991.CrossRefGoogle Scholar
- [7]K. Breitung. Crossing rates of Gaussian vector processes. In Transactions of the 11th Prague Conference on Information Theory, Statistical Decision Functions and Random Processes 199., volume I, pages 303–314, Prague, Czech. Rep., 1992. Academia.Google Scholar
- [8]K. Breitung. Asymptotic Approximations for Probability Integrals. Springer, New York, 1994. to appear.Google Scholar
- [9]K. Breitung and M. Hohenbichler. Asymptotic approximations for multivariate integrals with an application to multinormal probabilities. Journal of Multivariate Analysi., 30:80–97, 1989.CrossRefGoogle Scholar
- [10]K. Breitung and W.-D. Richter. An asymptotic expansion fot large deviation probabilities of Gaussian random vectors. Journal of Multivariate Analysi., 1993. submitted.Google Scholar
- [11]A.M. Freudenthal. Safety and the probability of structural failure. Trans-actions of the ASC., 121:1337–1397, 1956.Google Scholar
- [12]A.M. Hasofer and N.C. Lind. An exact and invariant first-order reliability format. Journal of the Engineering Mechanics Division ASC., 100(1):111–121, 1974.Google Scholar
- [13]M.A. Maes, K. Breitung, and D.J. Dupuis. Asymptotic importance sampling. Structural Safet., 1993. to appear.Google Scholar
- [14]M.A. Maes, K. Breitung, and P. Geyskens. Asymptotic importance sampling. In Y.K. Lin, editor, Probabilistic Mechanics and Structural and Geotechnical Reliability, Proceedings of the sixth specialty conferenc., pages 96–99. American Society of Civil Engineers, 1992.Google Scholar
- [15]W.-D. Richter. Laplace-Gauss integrals, Gaussian measure asymptotic behavior and probabilities of moderate deviations. Zeitschrift für Analysis und ihre Anwendunge., 4(3):257–267, 1985.Google Scholar
- [16]W.-D. Richter. Remarks on moderate deviations in the multidimensional central limit theorem. Mathematische Nachrichte., 122:167–173, 1985.CrossRefGoogle Scholar
- [17]R. Wong. Asymptotic Approximations of Integrals. Academic Press, San Diego, 1989.Google Scholar
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