Abstract
We treat the identifiability problem for mixtures involving power series distributions. Applying an idea of Sapatinas (Ann Inst Stat Math 47:447–459, 1995) we prove and elaborate that a mixture distribution is identifiable if a certain Stieltjes problem of moments has a unique solution while a non-uniqueness leads to a non- identifiable mixture. We describe explicitly models of identifiable mixtures and models of non-identifiable mixtures. Illustrative examples and comments on related questions are also given.
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References
Barndorff-Nielsen O. (1965) Identifiability of mixtures of exponential families. Journal of Mathematical Analysis and Applications 12: 115–121
Bingham N., Goldie C., Teugels J. (1989) Regular variation. Cambridge University Press, Cambridge
Böhning D., Patilea V. (2005) Asymptotic normality in mixtures of power series distributions. Scandinavian Journal of Statistics 32: 115–131
Everitt B.S., Hand D.J. (1981) Finite mixture distributions. Chapman & Hall, London
Feller, F. (1971). An introduction to probability theory and its applications (Vol. 2, 2nd ed.). New York: Wiley.
Gupta, A. K., Nguyen, T. T., Wang, Y., Wesolowski, J. (2001). Identifiability of modified power series mixtures via posterior means. Journal of Multivariate Analysis, 77, 163–174. Erratum, ibid, 80(2002), 189.
Gut A. (2002) On the moment problem. Bernoulli 8: 407–421
Holzmann H., Munk A., Stratmann B. (2004) Identifiability of finite mixtures—with applications to circular distributions. Sankhya 66: 440–449
Lin G.D. (1992) Characterizations of distributions via moments. Sankhya A54: 128–132
Lin G.D. (1993) Characterizations of the exponential distribution via the blocking time in a queueing system. Statistica Sinica 3: 577–581
Lin, G. D. (1997). On the moment problems. Statistics and Probability Letters, 35, 85–90. Erratum, ibid, 50 (2000), 205.
Lin G.D., Stoyanov J. (2002) On the moment determinacy of the distributions of compound geometric sums. Journal of Applied Probability 39: 545–554
Lindsay B.G. (1995) Mixture models: theory, geometry and applications. Institute of Mathematical Statistics, Hayward
Lüxmann-Ellinghaus U. (1987) On the identifiability of mixtures of infinitely divisible power series distributions. Statistics and Probability Letters 5: 375–378
McLachlan G., Peel D. (2000) Finite mixture models. Wiley, New York
Pakes A. (2001) Remarks on converse Carleman and Krein criteria for the classical moment problem. Journal of the Australian Mathematical Society 71: 81–104
Pakes A., Hung W.-L., Wu J.-W. (2001) Criteria for the unique determination of probability distributions by moments. Australian and New Zealand Journal of Statistics 43: 101–111
Papageorgiou H., Wesolowski J. (1997) Posterior mean identifies the prior distribution in NB and related models. Statistics and Probability Letters 36: 127–134
Prakasa Rao B.L.S. (1992) Identifiability in stochastic models: characterization of probability distributions. Academic Press, London
Prudnikov, A. P., Brychkov, Yu. A., Marichev, O. I. (1992). Integrals and series (Vol. 1). New York: Gordon & Breach.
Sapatinas T. (1995) Identifiability of mixtures of power-series distributions and related characterizations. Annals of the Institute of Statistical Mathematics 47: 447–459
Slud E.V. (1993) The moment problem for polynomial forms in normal random variables. Annals of Probability 21: 2200–2214
Stoyanov J. (1997) Counterexamples in probability (2nd ed). Wiley, Chichester
Stoyanov J. (2000) Krein condition in probabilistic moment problems. Bernoulli 6: 939–949
Stoyanov J. (2004) Stieltjes classes for moment-indeterminate probability distributions. Journal of Applied Probability 41A: 281–294
Stoyanov J., Tolmatz L. (2005) Method for constructing Stieltjes classes for M-indeterminate probability distributions. Applied Mathematics and Computation 165: 669–685
Tallis G.M. (1969) The identifiability of mixtures of distributions. Journal of Applied Probability 6: 389–398
Titterington D.M., Smith A.F.M., Makov U.E. (1985) Statistical analysis of finite mixtures distributions. Wiley, New York
Wesolowski J. (1995) Bivariate discrete measures via a power series conditional distributions and a regression function. Journal of Multivariate Analysis 55: 219–229
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Stoyanov, J., Lin, G.D. Mixtures of power series distributions: identifiability via uniqueness in problems of moments. Ann Inst Stat Math 63, 291–303 (2011). https://doi.org/10.1007/s10463-009-0221-9
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DOI: https://doi.org/10.1007/s10463-009-0221-9