On identifiability of mixtures of independent distribution laws

Mikhail Kovtun; Igor Akushevich; Anatoliy Yashin

ESAIM: Probability and Statistics (2014)

  • Volume: 18, page 207-232
  • ISSN: 1292-8100

Abstract

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We consider representations of a joint distribution law of a family of categorical random variables (i.e., a multivariate categorical variable) as a mixture of independent distribution laws (i.e. distribution laws according to which random variables are mutually independent). For infinite families of random variables, we describe a class of mixtures with identifiable mixing measure. This class is interesting from a practical point of view as well, as its structure clarifies principles of selecting a “good” finite family of random variables to be used in applied research. For finite families of random variables, the mixing measure is never identifiable; however, it always possesses a number of identifiable invariants, which provide substantial information regarding the distribution under consideration.

How to cite

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Kovtun, Mikhail, Akushevich, Igor, and Yashin, Anatoliy. "On identifiability of mixtures of independent distribution laws." ESAIM: Probability and Statistics 18 (2014): 207-232. <http://eudml.org/doc/274398>.

@article{Kovtun2014,
abstract = {We consider representations of a joint distribution law of a family of categorical random variables (i.e., a multivariate categorical variable) as a mixture of independent distribution laws (i.e. distribution laws according to which random variables are mutually independent). For infinite families of random variables, we describe a class of mixtures with identifiable mixing measure. This class is interesting from a practical point of view as well, as its structure clarifies principles of selecting a “good” finite family of random variables to be used in applied research. For finite families of random variables, the mixing measure is never identifiable; however, it always possesses a number of identifiable invariants, which provide substantial information regarding the distribution under consideration.},
author = {Kovtun, Mikhail, Akushevich, Igor, Yashin, Anatoliy},
journal = {ESAIM: Probability and Statistics},
keywords = {latent structure analysis; mixed distributions; identifiability; moment problem; mixed distribution},
language = {eng},
pages = {207-232},
publisher = {EDP-Sciences},
title = {On identifiability of mixtures of independent distribution laws},
url = {http://eudml.org/doc/274398},
volume = {18},
year = {2014},
}

TY - JOUR
AU - Kovtun, Mikhail
AU - Akushevich, Igor
AU - Yashin, Anatoliy
TI - On identifiability of mixtures of independent distribution laws
JO - ESAIM: Probability and Statistics
PY - 2014
PB - EDP-Sciences
VL - 18
SP - 207
EP - 232
AB - We consider representations of a joint distribution law of a family of categorical random variables (i.e., a multivariate categorical variable) as a mixture of independent distribution laws (i.e. distribution laws according to which random variables are mutually independent). For infinite families of random variables, we describe a class of mixtures with identifiable mixing measure. This class is interesting from a practical point of view as well, as its structure clarifies principles of selecting a “good” finite family of random variables to be used in applied research. For finite families of random variables, the mixing measure is never identifiable; however, it always possesses a number of identifiable invariants, which provide substantial information regarding the distribution under consideration.
LA - eng
KW - latent structure analysis; mixed distributions; identifiability; moment problem; mixed distribution
UR - http://eudml.org/doc/274398
ER -

References

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  9. [9] J.C. Oxtoby, Measure and Category. Number 2 in Graduate Texts in Mathematics. Springer-Verlag, New York, 2nd edition (1980). ISBN 0-378-90508-1. Zbl0435.28011MR584443
  10. [10] A.N. Shiryaev, Probability. Moscow, Russia: MCCSE, 3rd edition (2004). In Russian. 
  11. [11] G.M. TallisandP. Chesson, Identifiability of mixtures. J. Austral. Math. Soc. Ser. A 32 (1982) 339–348. ISSN 0263-6115. Zbl0491.62012MR652411
  12. [12] H. Teicher, On the mixture of distributions. Ann. Math. Stat.31 (1960) 55–73. Zbl0107.13501MR121825
  13. [13] H. Teicher, Identifiability of mixtures. Ann. Math. Stat.32 (1961) 244–248. Zbl0146.39302MR120677
  14. [14] H. Teicher, Identifiability of finite mixtures. Ann. Math. Stat.34 (1963) 1265–1269. Zbl0137.12704MR155376

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