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On structural approximating multivariate discrete probability distributions. (English) Zbl 0547.62034
The purpose of structural approximating is to develop a flexible parametric model applicable to estimating various types of probability distributions. The approach combines the idea of product approximating with the concept of finite mixtures. To optimize mixtures of product approximations the maximum-likelihood principle is used. An efficient numerical solution is enabled by a convergent iterative procedure. A numerical example previously used by other authors is included to compare different structural approximations. (Author’s summary)
Reviewer: J.Panaretos

##### MSC:
 62H05 Characterization and structure theory for multivariate probability distributions; copulas 65C99 Probabilistic methods, stochastic differential equations 62H12 Estimation in multivariate analysis 62E20 Asymptotic distribution theory in statistics
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##### References:
 [1] D. T. Brown: A note on approximations to discrete probability distributions. Inform. and Control 2 (1959), 4, 386-392. · Zbl 0117.14804 · doi:10.1016/S0019-9958(59)80016-4 [2] C. K. Chow: Tree dependence in normal distributions. International Symposium on Information Theory, Nordwijk, The Netherlands 1970. [3] C. K. Chow, C. N. Liu: Approximating discrete probability distributions with dependence trees. IEEE Trans. Inform. Theory IT-14 (1968), 3, 462-467. · Zbl 0165.22305 · doi:10.1109/TIT.1968.1054142 [4] C. K. Chow, T. J. Wagner: Consistency of an estimate of tree-dependent probability distribution. IEEE Trans. Inform. Theory IT-19 (1973), 5, 369-371. [5] W. E. Deming, F. F. Stephan: On a least squares adjustement of a sampled frequency table when the expected marginal totals are known. Ann. Math. Statist. 77 (1940), 427-444. · Zbl 0024.05502 · doi:10.1214/aoms/1177731829 [6] W. A. Gibson: Three multivariate models: factor analysis, latent structure analysis and latent profile analysis. Psychometrika 24 (1959), 229 - 252. · Zbl 0117.15001 · doi:10.1007/BF02289845 [7] J. Grim: On estimation of multivariate probability density functions for situation recognition in large scale systems. Proceedings of Third Formator Symposium (J. Benes, L. Bakule, Academia, Prague 1979. · Zbl 0485.93006 [8] J. Grim: On numerical evaluation of maximum-likelihood estimates for finite mixtures of distributions. Kybernetika 18 (1982), 3, 173-190. · Zbl 0489.62028 · eudml:27287 [9] J. Grim: Application of finite mixtures to multivariate statistical pattern recognition. Proceedings of DIANA Conference held in Liblice near Prague, September 27 - October 1, 1982. [10] C. T. Ireland, S. Kullback: Contingency tables with given marginals. Biometrika 55 (1968), 179-188. · Zbl 0155.26701 · doi:10.1093/biomet/55.1.179 [11] А. Д. Юдин: Об информативных структурах многомерных случайных величин. (On information structures of multidimensional random variables.) Известия АН СССР - Тєхническая кибернетика (1977), 6, 135-144. · Zbl 1170.01341 [12] H. H. Ku, S. Kullback: Approximating discrete probability distributions. IEEE Trans. Inform. Theory IT-15 (1969), 444-447. · Zbl 0174.23202 · doi:10.1109/TIT.1969.1054336 [13] S. Kullback: Probability densities with given marginals. Ann. Math. Statist. 39 (1968), 4, 1236-1243. · Zbl 0165.20303 · doi:10.1214/aoms/1177698249 [14] P. F. Lazarsfeld, N. W. Henry: Latent structure analysis. Houghton Mifflin, Boston 1968. · Zbl 0182.52201 [15] P. M. Lewis: Approximating probability distributions to reduce storage requirements. Inform. and Control 2 (1959), 214-225. · Zbl 0095.32602 · doi:10.1016/S0019-9958(59)90207-4 [16] A. Perez: $$\epsilon$$-admissible simplification of the dependence structure of a set random variables. Kybernetika 13 (1977), 6, 439-449. · Zbl 0382.62003 · eudml:28225 [17] R. C. Prim: Shortest connection networks and some generalizations. Bell System Tech. J. 36 (1957), 1389-1401. [18] F. F. Stephan: An iterative method of adjusting sample frequency tables when expected marginal totals are known. Ann. Math. Statist. 13 (1942), 166- 178. · Zbl 0060.31505 · doi:10.1214/aoms/1177731604
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