Multivariate Mixtures of Normal Distributions: Properties, Random Vector Generation, Fitting, and as Models of Market Daily Changes

References

• Alexander G, Baptista A (2002) Economic implications of using a mean-var model for portfolio selection: A comparison with mean-variance analysis. J. Econom. Dynam. Control 26:1159–1193.Crossref, Google Scholar
• Atkinson KE (1989) An Introduction to Numerical Analysis, 2nd ed. (John Wiley & Sons, New York).Google Scholar
• Behboodian J (1970) On a mixture of normal distributions. Biometrika 57:215–217.Crossref, Google Scholar
• Biller B, Ghosh S (2006) Multivariate input processes. Nelson BL, Henderson SG, eds. Handbook in Operations Research and Management Science: Simulation (Elsevier Science, Amsterdam),123–154.Google Scholar
• Biller B, Nelson BL (2003) Modeling and generating multivariate time-series input processes using a vector autoregressive technique. ACM Trans. Modeling Comput. Simulation 13:211–237.Crossref, Google Scholar
• Clark PK (1973) A subordinated stochastic process model with finite variance for speculative prices. Econometrica 41:135–155.Crossref, Google Scholar
• Cohen AC (1967) Estimation in mixtures of two normal distributions. Technometrika 9:15–28.Crossref, Google Scholar
• Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Statist. Soc. B39:1–38.Google Scholar
• Dowd K (1998) Beyond Value at Risk: The New Science of Risk Management (John Wiley & Sons, Chichester, UK).Google Scholar
• Duffie D, Pan J (1997) An overview of value at risk. J. Derivatives 4:7–49.Crossref, Google Scholar
• Fishman GS (1973) Concepts and Methods in Discrete Event Digital Simulation (John Wiley & Sons, New York).Google Scholar
• Ghosh S, Henderson S (2003) Behavior of the NORTA method for correlated random vector generation as the dimension increases. ACM Trans. Modeling Comput. Simulation 13:276–294.Crossref, Google Scholar
• Glasserman P, Heidelberger P, Shahabuddin P (2002) Portfolio value-at-risk with heavy-tailed risk factors. Math. Finance 12:239–269.Crossref, Google Scholar
• Hamerly G, Elkan C (2003) Learning the k in k-means. Thrun S, Saul LK, Scholkopf B, eds. Proc. Adv. Neural Inform. Processing Systems, Vol. 16 (MIT Press, Cambridge, MA), 281–288.Google Scholar
• Hamilton JD (1991) A quasi-Bayesian approach to estimating parameters for mixtures of normal distributions. J. Bus. Econom. Statist. 9:27–39.Crossref, Google Scholar
• Hathaway RJ (1985) A constrained formulation of maximum-likelihood estimation for normal mixture distributions. Ann. Statist. 13:795–800.Crossref, Google Scholar
• Hull J, White A (1998) Value-at-risk when daily changes in market variables are not normally distributed. J. Derivatives 5:9–19.Crossref, Google Scholar
• Johnson ME (1987) Multivariate Statistical Simulation (John Wiley & Sons, New York).Crossref, Google Scholar
• Jorion P (1997) Value at Risk: The New Benchmark for Controlling Market Risk (McGraw-Hill, New York).Google Scholar
• Law AM, Kelton WD (2000) Simulation Modeling and Analysis, 3rd ed. (McGraw-Hill, New York).Google Scholar
• Li DX (1999) Value at risk based on volatility, skewness and kurtosis. Technical report, RiskMetrics Group, New York.Google Scholar
• Li ST, Hammond JL (1975) Generation of pseudorandom numbers with specified univariate distributions and correlation coefficients. IEEE Trans. Systems, Man Cybernetics 5:557–561.Crossref, Google Scholar
• Mardia KV (1970) A translation family of bivariate distributions and Fréchet’s bounds. Sankhya A32:119–122.Google Scholar
• McLachlan GJ, Peel D (1998) Mixfit: An algorithm for the automatic fitting and testing of normal mixture models. Thrun S, Saul LK, Scholkopf B, eds. Proc. Fourteenth Internat. Conf. Pattern Recognition (IEEE Computer Society, Los Alamitos, CA), 553–557.Crossref, Google Scholar
• McLachlan GJ, Peel D (2000) Finite Mixture Models (John Wiley & Sons, New York).Crossref, Google Scholar
• Morgan JP (1995) Riskmetrics™-technical documentation, release 1–3. Technical report, J.P. Morgan, New York.Google Scholar
• Pearson K (1894) Contributions to the theory of mathematical evolution. Philosophical Trans. Roy. Soc. London A185:71–110.Crossref, Google Scholar
• Quandt RE, Ramsey JB (1978) Estimating mixture of normal distributions and switching regressions. J. Amer. Statist. Assoc. 73:730–738.Crossref, Google Scholar
• Rubinstein RY (1981) Simulation and the Monte Carlo Method (John Wiley & Sons, New York).Crossref, Google Scholar
• Schmeiser BW (1991) IE 581 Lecture Notes: Three-step method. School of Industrial Engineering, Purdue University, West Lafayette, IN.Google Scholar
• Titterington DM, Smith AFM, Makov UE (1985) Statistical Analysis of Finite Mixture Distribution (John Wiley & Sons, New York).Google Scholar
• Venkataraman S (1997) Value at risk for a mixture of normal distributions: The use of quasi-Bayesian estimation techniques. Econom. Perspect. March/April:2–13.Google Scholar
• Wang J (2000) Mean-variance-var based portfolio optimization. Technical report, Department of Mathematics and Computer Science, Valdosta State University, GA.Google Scholar
• Wang J, Liu C (2006) Generating multivariate mixture of normal distributions using a modified Cholesky decomposition. Perrone LF, Wieland FP, Liu J, Lawson BG, Nicol DM, Fujimoto RM, eds. Proc. 2006 Winter Simulation Conf., Monterey, CA, 342–347.Crossref, Google Scholar
• Wilson TC (1993) Infinite wisdom. Risk 6:37–45.Google Scholar
• Wilson TC (1998) Value at risk. Alexander C, ed. Risk Management and Analysis, Vol. 1 (John Wiley & Sons, New York), 61–124.Google Scholar
• Wirjanto T, Xu D (2009) The applications of mixtures of normal distributions in empirical finance: A selected survey. Technical report, University of Waterloo, Ontario, Canada.Google Scholar
• Wu CFJ (1983) On the convergence properties of the EM algorithm. Ann. Statist. 11:95–103.Crossref, Google Scholar
• Xu L, Jordan MI (1996) On the convergence properties of the EM algorithm for Gaussian mixtures. Neural Comput. 8:129–151.Crossref, Google Scholar
• Zangari P (1996) An improved methodology for measuring var. Technical report, Reuters/JP Morgan. RiskMetrics™ Monitor, New York.Google Scholar