Develop computer programs for multivariate normal probability bounds by Rack witz’s procedure and Ditlevsen’s procedure, respectively.

References

Abramowitz, M., and Stegun, I.A., eds. (1972). Handbook of Mathematical Functions with Formulas,

Graphs and Mathematical Tables, 9th ed., Dover Publications, New York.

Ang, A. H. S., and Tang, W. H. (1975). Probability Concepts in Engineering Planning and Design, Vol. I: Basic Principles, John Wiley and Sons, New York.

Blank, L. (1980). Statistical Procedures for Engineering, Management, and Science, McGraw-Hill, New York.

Cong, S. Z., and Y. B. Xu (1987). “The effect of discharge measurement error in flood frequency analysis,” Journal of Hydrology, 96:237–254.

Consul, P. C. (1989). Generalized Poisson Distributions: Properties and Application, Marcel Dekker, New York.

Consul, P. C., and Jain, G. C. (1973). A generalization of Poisson distribution: Properties and ap- plication, Technometrics, 15:791–799.

DeGroot, M. H. (1975). Probability and Statistics, Addison-Wesley, Reading, MA.

Der Kiureghian, A., and Liu, P. L. (1986). Structural reliability under incomplete probability dis- tribution, Journal of Engineering Mechanics, ASCE, 112(1):85–104.

Devore, J. L. (1987). Probability and Statistics for Engineering and Sciences, 2d ed., Brooks/Cole Publishing, Monterey, CA.

Ditlevsen, O. (1979). Narrow reliability bounds for structural systems, Journal of Structural Me-

chanics, 7(4):435–451.

Ditlevsen, O. (1982). System reliability bounding by conditioning, Journal of Engineering Mechan-

ics, ASCE, 108(5):708–718.

Ditlevsen, O. (1984). Taylor expansion of series system reliability, Journal of Engineering Mechan-

ics, ASCE, 110(2):293–307.

Donnelly, T. G. (1973). Algorithm 462: Bivariate normal distribution, Communications, Association for Computing Machinery, 16:638.

Dowson, D. C., and Wragg, A. (1993). Maximum entropy distribution having prescribed first and second moments, IEEE Transaction on Information Theory, 19(9):689–693.

Drane, J. W., Cao, S., Wang, L., and Postelnicu, T. (1993). Limiting forms of probability mass function via recurrence formulas, The American Statistician, 47(4):269–274.

Dudewicz, E. (1976). Introduction to Statistics and Probability, Holt, Rinehart, and Winston, New York.

Farlie, D. J. G. (1960). The performance of some correlation coefficients for a general bivariate distribution, Biometrika, 47:307–323.

Fisher, R. A., and Tippett, L. H. C. (1928). Limiting forms of the frequency distribution of the largest or smallest member of a sample, Proceding of the Cambridge Philosphical Society, 24:180–190. Greenwood, J. A., Landwehr, J. M., Matalas, N. C., and Wallis, J. R. (1979). Probability-weighted moments: Definitions and relation to parameters of several distribution expressible in inverse form, Water Resources Research, 15(6):1049–1054.

Gnedenko, B. V. (1943). Sur la distribution limite du terme maximum d’une serie aleatoire, Annals

of Mathematics, 44:423–453.

Gumbel, E. J. (1958). Statistics of Extremes, Columbia University Press, New York.

Haan, C. T. (1977). Statistical Methods in Hydrology, Iowa State University Press, Ames, IA. Harr, M. E. (1987). Reliability-Based Design in Civil Engineering, McGraw-Hill, New York.

Hosking, J. R. M. (1986). The theory of probability-weighted moments, IBM Research Report, No. 12210, October.

Hosking, J. R. M. (1990). L-moments: Analysis and estimation of distributions using linear combi- nations of order statistics, Journal of Royal Statistical Society, Series B, 52(1):105–124. Hosking, J. R. M. (1991). Approximations for use in constructing L-moment ratio diagram, IBM

Research Report No. 16635, T. J. Watson Research Center, Yorktown Heights, NY.

Hutchinson, T. P., and Lai, C. D. (1990). Continuous Bivariate Distributions, Emphasizing Appli-

cations, Rumsby Scientific Publishing, Adelaide, South Australia.

Johnson, M. E. (1987). Multivariate Statistical Simulation, John Wiley and Sons, New York. Johnson, N. L., and Kotz, S. (1972). Distributions in Statistics: Continuous Univariate Distributions,

Vol. 2. John Wiley and Sons, New York.

Johnson, N. L., and Kotz, S. (1976). Distributions in Statistics: Continuous Multivariate Distribu-

tions, John Wiley and Sons, Inc., New York, NY.

Kite, G. W. (1988). Frequency and Risk Analyses in Hydrology, Water Resources Publications, Littleton, CO.

Leadbetter, M. R., Lindgren, G., and Rootzen, H. (1983). Extremes and Related Properties of Random

Sequences and Processes, Springer-Verlag, New York.

Leemis, L. M. (1986). Relationships among common univariate distributions, The American Statis-

tician, 40(2):143–146.

Liu, P. L., and Der Kiureghian, A. (1986). Multivariate distribution models with prescribed marginals and covariances, Probabilistic Engineering Mechanics, 1(2):105–112.

Mardia, K. V. (1970a). Families of Bivariate Distributions, Griffin, London.

Mardia, K. V. (1970b). A translation family of bivariate distributions and Frechet’s bounds,

Sankhya, Series A, 32:119–121.

Mays, L. W., and Tung, Y. K. (1992). Hydrosystems Engineering and Management, McGraw-Hill, New York.

Morgenstern, D. (1956). Einfache Beispiele zweidimensionaler Verteilungen, Mitteilingsblatt fur

Mathematische Statistik, 8:234–235.

Nataf, A. (1962). Determination des distributions dont les marges sont donnees, Comptes Rendus

de l’Academie des Sciences, Paris, 225:42–43.

Rackwitz, R. (1978). Close bounds for the reliability of structural sytems, Berichte zur Zuverlas-

sigkeitstheorie der Bauwerke, SFB 96, Heft 29/1978, LKI, Technische Universitat Munchen.

Rao, A. R., and Hamed, K. H. (2000). Flood Frequency Analysis. CRC Press, Boca Raton, FL. Royston, P. (1992). Which measures of skewness and kurtosis are best? Statistics in Medicine,

11:333–343.

Ruben, H. (1964). An asymptotic expansion for the multivariate normal distribution and Mill’s ratio, Journal of Research National Bureau of Standards, 68B(1):3–11.

Sartor, J. D., and Boyd, G. B. (1972). Water pollution aspects of street surface contaminants. Report No. EPA-R2-72-081, U.S. Environmental Protection Agency, Washington.

Shen, H. W., and Bryson, M. (1979). Impact of extremal distributions on analysis of maximum loading, in Reliability in Water Resources Engineering, ed. by E. A. McBean, K. W. Hipel, and T. E. Unny, Water Resources Publications, Littleton, CO.

Stedinger, J. R., Vogel, R. M., and Foufoula-Georgoiu, E. (1993). Frequency analysis of extreme events, in Handbook of Hydrology, ed. by D. R. Maidment, McGraw-Hill, New York.

Stuart, A., and Ord, J. K. (1987). Kendall’s Advanced Theory of Statistics, Vol. 1: Distribution

Theory, 5th ed., Oxford University Press, New York.

Tietjen, G. (1994). Recursive schemes for calculating cumulative binomial and Poisson probabilities,

The American Statistician, 48(2):136–137.

Tung, Y. K., and Yen, B. C. (2005). Hydrosystems Engineering Uncertainty Analysis, McGraw-Hill, New York.

U.S. Water Resources Council (now called the Interagency Advisory Committee on Water Data) (1982). Guideline in Determining Flood Flow Frequency, Bulletin 17B; available from Officie of Water Data Coordination, U.S. Geological Survey, Reston, VA.

Vale, C. D., and Maurelli, V. A. (1983). Simulating multivariate nonnormal distributions, Psychome-

trika, 48(3):65–471.

Vedder, J. D. (1995). An invertible approximation to the normal distribution function, Computa-

tional Statistics and Data Analysis, 16(2):119–123.

Wahl, K. L., and Rankl, J. G. (1993). A potential problem with mean dimensionless hydrographs at ungaged sites, in Proceedings, International Symposium on Engineering Hydrology, San Fran- cisco, CA, July 26–30.

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