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.

Classification: 60E99

Keywords: latent structure analysis, mixed distributions, identifiability, moment problem

@article{PS_2014__18__207_0, author = {Kovtun, Mikhail and Akushevich, Igor and Yashin, Anatoliy}, title = {On identifiability of mixtures of independent distribution laws}, journal = {ESAIM: Probability and Statistics}, publisher = {EDP-Sciences}, volume = {18}, year = {2014}, pages = {207-232}, doi = {10.1051/ps/2011166}, mrnumber = {3230875}, language = {en}, url = {http://www.numdam.org/item/PS_2014__18__207_0} }

Kovtun, Mikhail; Akushevich, Igor; Yashin, Anatoliy. On identifiability of mixtures of independent distribution laws. ESAIM: Probability and Statistics, Volume 18 (2014) pp. 207-232. doi : 10.1051/ps/2011166. http://www.numdam.org/item/PS_2014__18__207_0/

[1] Linear latent structure analysis and modeling of multiple categorical variables. Comput. Math. Methods Medicine 10 (2009) 203-218. | MR 2566934 | Zbl pre05935210

, , and ,[2] Identifiability of mixtures of exponential families. J. Math. Anal. Appl. 12 (1965) 115-121. | MR 183058 | Zbl 0138.12105

,[3] Probabilities and Potential, vol. I. North-Holland Publishing Co., Amsterdam (1978). | MR 521810 | Zbl 0494.60001

and ,[4] Linear Operators, vol. I. Interscience Publishers, Inc., New York (1958). | MR 117523 | Zbl 0084.10402

, and ,[5] Finite Mixture Distributions. Monographs on Applied Probability and Statistics. Chapman and Hall, London (1981). | MR 624267 | Zbl 0466.62018

, and ,[6] Probability Theory and Stochastic Processes with Applications. Overseas Press (2009). ISBN 81-89938-40-1.

,[7] Elements of the theory of functions and functional analysis. Moscow, Russia, Science, 3rd edition (1972). In Russian. | Zbl 0235.46001

and ,[8] Linear latent structure analysis: Mixture distribution models with linear constraints. Statist. Methodology 4 (2007) 90-110. | MR 2339011 | Zbl 1248.62115

, , and ,[9] Measure and Category. Number 2 in Graduate Texts in Mathematics. Springer-Verlag, New York, 2nd edition (1980). ISBN 0-378-90508-1. | MR 584443 | Zbl 0435.28011

,[10] Probability. Moscow, Russia: MCCSE, 3rd edition (2004). In Russian.

,[11] P. Chesson, Identifiability of mixtures. J. Austral. Math. Soc. Ser. A 32 (1982) 339-348. ISSN 0263-6115. | MR 652411 | Zbl 0491.62012

[12] On the mixture of distributions. Ann. Math. Stat. 31 (1960) 55-73. | MR 121825 | Zbl 0107.13501

,[13] Identifiability of mixtures. Ann. Math. Stat. 32 (1961) 244-248. | MR 120677 | Zbl 0146.39302

,[14] Identifiability of finite mixtures. Ann. Math. Stat. 34 (1963) 1265-1269. | MR 155376 | Zbl 0137.12704

,