On numerical evaluation of maximum-likelihood estimates for finite mixtures of distributions

Grim, Jiří

Kybernetika, Tome 18 (1982) no. 3, pp. 173-190

Détail

Classification : 62F10, 62F12, 65C99

Bibliographie

[1] C. A. Aйвазян З. И. Бежаева O. В. Cтароверов: Kлассификация многомерных наблюдений. (Classification of Multivariate Observations). Статистика, Mocквa 1974. | Zbl

[2] H. H. Aпраушева: Алгоритм расщепления смеси нормальных классов. (Algorithm for resolution of a mixture of normal classes). C6. Программы и алгоритмы (1976), 68. | Zbl

[3] H. H. Aпраушева: Определение числа классов в задачах классификации I. (Determination of the number of classes in classification problems I). Известия AH CCCP - Teх. kибернетика (1981), 3, 71-77. | MR | Zbl

[4] J. Behboodian: On a mixture of normal distributions. Biometrika 57 (1970), 1, 215-217. | Zbl

[5] C. G. Bhattacharya: A simple method of resolution of a distribution into Gaussian components. Biometrics 23 (1967), 115-137.

[6] W. R. Blischke: Moment estimators for the parameters of a mixture of two binomial distributions. Ann. Math. Statist. 33 (1962), 2, 444-454. | MR | Zbl

[7] W. R. Blischke: Mixtures of distributions. In: Classical and Contagious Discrete Distributions (G. P. Patil, ed.), Pergamon Press, New York 1963.

[8] W. R. Blischke: Estimating the parameters of mixtures of binomial distributions. J. Amer. Statist. Assoc. 59 (1964), 306, 510-528. | MR | Zbl

[9] C. Bürrau: The half-invariants of the sum of two typical laws of errors with an application to the problem of dissecting a frequency curve into components. Scand. Actuar. J. 77 (1934), 1, 1-5.

[10] C. V. L. Charlier: Researches into the theory of probability. Meddelanden fran Lunds Astron. Observ. (1906) Sec. 2, Bd. 1.

[11] A. C. Cohen: Discussion of "Estimation of parameters for a mixture of normal distributions" by Victor Hasselblad. Technometrics 8 (1966), 3, 445-446. | MR

[12] A. C. Cohen: Estimation in mixtures of two normal distributions. Technometrics 9 (1967), 15-28. | MR | Zbl

[13] P. W. Cooper: Some topics on nonsupervised adaptive detection for multivariate normal distributions. In: Computer and Information Sciences - II. (J. T. Tou, ed.), Academic Press, New York 1967. | Zbl

[14] N. E. Day: Estimating the components of a mixture of normal distributions. Biometrika 56 (1969), 463-474. | MR | Zbl

[15] N. P. Dick D. C. Bowden: Maximum likelihood estimation for mixtures of two normal distributions. Biometrics 29 (1973), 4, 781 - 790.

[16] G. Doetsch: Zerlegung einer Funktion in Gaussche Fehlerkurven und zeitliche Zurückverfolgung eines Temperaturzustandes. Math. Z. 41 (1936), 283 - 318. | MR

[17] R. O. Duda P. E. Hart: Pattern Classification and Scene Analysis. John Wiley, New York-London 1973.

[18] E. B. Fowlkes: Some methods for studying the mixture of two normal (lognormal) distributions. J. Amer. Statist. Assoc. 74 (1979), 367, 561-575. | Zbl

[19] J. G. Fryer C. A. Roberston: A comparison of some methods of estimating mixed normal distributions. Biometrika 59 (1972), 639-648. | MR

[20] N. T. Gridgeman: A comparison of two methods of analysis of normal distributions. Technometrics 12 (1970), 4, 832-833.

[21] J. Grim: Metody shlukové analýzy a jejich využití při zpětnovazebním řízení velkých systému. (Methods of cluster analysis and their application for feedback control of large systems). Dissertation, Institute of Information Theory and Automation, Prague 1979.

[22] J. Grim: An algorithm for maximizing a finite sum of positive functions and its application to cluster analysis. Problems of Control and Information Theory 10 (1981), 6, 427-437. | MR | Zbl

[23] E. J. Gumbel: La dissection d'une repartition. Annales de l'Université de Lyon 3 (1939), 39-51. | Zbl

[24] A. K. Gupta T. Miyawaki: On uniform mixture model. Biometrical J. 20 (1978), 631 - 637. | MR

[25] L. F. Guseman J. R. Walton: Methods for estimating proportions of convex combinations of normals using linear feature selection. Comm. Statist. A - Theory Methods A7 (1978), 1439-1450.

[26] V. Hasselblad: Estimation of parameters for a mixture of normal distributions. Technometrics 8 (1966), 431-444. | MR

[27] V. Hasselblad: Finite mixtures of distributions from the exponential family. Ph. D. Dissertation University of California, Los Angeles 1967.

[28] V. Hasselblad: Estimation of finite mixtures of distributions from the exponential family. J. Amer. Statist. Assoc. 64 (1969), 328, 1459-1471.

[29] B. M. Hill: Information for estimating the proportions in mixtures of exponential and normal distributions. J. Amer. Statist. Assoc. 58 (1963), 918-932. | MR

[30] D. W. Hosmer: A comparison of iterative maximum likelihood estimates of the parameters of a mixture of two normal distributions under three different types of samples. Biometrics 29 (1973), 761-770.

[31] D. W. Hosmer: A use of mixtures of two normal distributions in a classification problem. J. Statist. Comput. Simulation 6 (1978), 384, 281-294. | Zbl

[32] D. W. Hosmer N. P. Dick: Information and mixtures of two normal distributions. J. Statist. Comput. Simulation 6 (1977), 137-148.

[33] O. К. Исаенко К. Ю. Ypбax: Разделение смесей распределений вероятностей на их составляюшие. (Decomposition of mixtures of probability distributions into their components). Теория вероятностей и математическая статистика, теор. кибернетика, том 13, 37-58, ВИНИТИ, Mocквa 1976.

[34] I. R. James: Estimation of the mixing proportion in a mixture of two normal distributions from simple rapid measurements. Biometrics 34 (1978), 2, 265-275. | Zbl

[35] E. John: Bayesian estimation of mixture distributions. Ann. Math. Statist, 39 (1968), 4, 1289-1302. | MR

[36] B. K. Kale: On the solution of likelihood equations by iteration processes: The multiparametric case. Biometrika 49 (1962), 479-486. | MR | Zbl

[37] R. Kanno: Estimation of parameters for a mixture of two normal distributions. Rep. Statist. Appl. Res. JUSE 22 (1975), 4, 1-15. | MR | Zbl

[38] S. Kullback: An information-theoretic derivation of certain limit relations for a stationary Markov Chain. SIAM J. Control 4 (1966), 3, 454-459. | MR | Zbl

[39] S. Kullback: Information Theory and Statistics. Wiley, New York-Dover 1968. | MR

[40] P. D. M. Macdonald: Estimation of finite distribution mixtures. In: Applied Statistics (R. P. Gupta, ed.), North-Holland 1975. | MR | Zbl

[41] P. Medgyessy: Decomposition of Superpositions of Density Functions and Discrete Distributions. Akadémiai Kiadó, Budapest 1977. | MR | Zbl

[42] G. Meeden: Bayes estimation of the mixing distributions, the discrete case. Ann. Math. Statist. 43 (1972), 6, 1993-1999. | MR

[43] W. Molenaar: Survey of estimation methods for a mixture of two normal distributions. Statist. Neerlandica 19 (1965), 4, 249-265. | MR

[44] G. D. Murray D. M. Titterington: Estimation problems with data from a mixture. Appl. Statist. 27(1978), 3, 325-334.

[45] K. Pearson: Contributions to the mathematical theory of evolution 1: Dissection of frequency curves. Philos. Trans. Roy. Soc. London Ser. A 185 (1894), 71-110.

[46] B. C. Peters W. A. Coberly: The numerical evaluation of the maximum-likelihood estimate of mixture proportions. Comm. Statist. A - Theory Methods A5 (1976), 12, 1127-1135. | MR

[47] B. C. Peters H. F. Walker: An iterative procedure for obtaining maximum-likelihood estimates of the parameters for a mixture of normal distributions. SIAM J. Appl. Math. 35 (1978), 2, 362-378. | MR

[48] B. C. Peters H. F. Walker: The numerical evaluation of the maximum-likelihood estimate of a subset of mixture proportions. SIAM J. Appl. Math. 35 (1978), 3,447-452. | MR

[49] J. G. Postaire C. P. A. Vasseur: An approximate solution to normal mixture identification with application to unsupervised pattern classification. IEEE Trans, on Pattern Analysis & Machine Intelligence PAMI-3 (1981), 2, 163-179.

[50] R. E. Quandt J. B. Ramsey: Estimating mixtures of normal distributions and switching regressions. J. Amer. Statist. Assoc. 73 (1978), 364, 730-752. | MR

[51] C. R. Rao: Advanced Statistical Methods in Biometric Research. John Wiley and Sons, New York 1952. | MR | Zbl

[52] P. R. Rider: The method of moments applied to a mixture of two exponential distributions. Ann. Math. Statist. 32 (1961), 1, 143-147. | MR | Zbl

[53] W. Schilling: A frequency distribution represented as the sum of two Poisson distributions. J. Amer. Statist. Assoc. 42 (1947), 407-424. | MR

[54] D. F. Stanat: Unsupervised learning of mixtures of probability functions. In: Pattern Recognition (L. Kanal, ed.), Thompson Book Co., Washington D. C. 1968, 357-389.

[55] B. Stromgren: Tables and diagrams for dissecting a frequency curve into components by the half-invariant method. Scand. Actuar. J. 17 (1934), 1, 7-54.

[56] M. И. Шлезингер: Взаимосвязъ обучения и самообучения в разпознавании образов. (Relation between learning and self-learning in pattern recognition). Кивернетика (Kиев) (1968), 2, 81-88. | Zbl

[57] W. Y. Tan W. C. Chang: Some comparisons of the method of moments and the method of maximum likelihood in estimating parameters of a mixture of two normal densities. J. Amer. Statist. Assoc. (57(1972), 339, 702-708.

[58] H. F. Walker: Estimating the proportions of two populations in a mixture using linear maps. Comm. Statist. A - Theory Methods A9 (1980), 8, 837-849. | MR | Zbl

[59] J. H. Wolfe: A computer program for the maximum likelihood analysis of types. (Technical Bulletin 65-15), U.S. Naval Personnel Research Activity, San Diego 1965.

[60] J. H. Wolfe: NORMIX: computational methods for estimating the parameters of multivariate normal mixtures of distributions. (Research Memorandum SRM 68-2), U.S. Naval Personnel Research Activity, San Diego 1967.

[61] J. H. Wolfe: Pattern clustering by multivariate mixture analysis. Multivariate Behavioral Research 5 (1970), July, 329-350.

[62] S. J. Yakowitz: Unsupervised learning and the identification of finite mixtures. IEEE Trans. Inform. Theory IT- 16 (1970), 5, 330-338. | Zbl

[63] T. Y. Young G. Coraluppi: Stochastic estimation of a mixture of normal density functions using an information criterion. IEEE Trans. Inform. Theory IT-16 (1970), 258-263.