Journal of Classification

, Volume 25, Issue 1, pp 113–123 | Cite as

A Simple Identification Proof for a Mixture of Two Univariate Normal Distributions

  • Erik MeijerEmail author
  • Jelmer Y. Ypma


A simple proof of the identification of a mixture of two univariate normal distributions is given. The proof is based on the equivalence of local identification with positive definiteness of the information matrix and the equivalence of the latter to a condition on the score vector that is easily checked for this model. Two extensions using the same line of proof are also given.


Finite mixtures Information matrix Exponential family Mixture regression 


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  1. AL-HUSSAINI, E.K. and AHMAD, K. E.-D. (1981), “On the Identifiability of Finite Mixtures of Distributions”, IEEE Transactions on Information Theory, 27, 664–668.zbMATHCrossRefGoogle Scholar
  2. BARNDORFF-NIELSEN, O. (1965), “Identifiability of Mixtures of Exponential Families”, Journal of Mathematical Analysis and Applications, 12, 115–121.zbMATHCrossRefMathSciNetGoogle Scholar
  3. BEKKER, P. and WANSBEEK, T. (2001), “Identification in Parametric Models”, in A Companion to Theoretical Econometrics, ed. B. H. Baltagi, Malden, MA: Blackwell, pp. 144–161.Google Scholar
  4. CHANDRA, S. (1977), “On the Mixtures of Probability Distributions”, Scandinavian Journal of Statistics, 4, 105–112.Google Scholar
  5. EVERITT, B.S. and HAND, D.J. (1981), Finite Mixture Distributions, London: Chapman and Hall.zbMATHGoogle Scholar
  6. FERRARI, S.L.P., CORDEIRO, G.M., URIBE-OPAZO, M.A., and CRIBARI-NETO, F. (1996), “Improved Score Tests for One-Parameter Exponential Family Models”, Statistics and Probability Letters, 30, 61–71.zbMATHCrossRefMathSciNetGoogle Scholar
  7. HENNIG, C. (2000), “Identifiability of Models for Clusterwise Linear Regression”, Journal of Classification, 17, 273–296.zbMATHCrossRefMathSciNetGoogle Scholar
  8. HILL, B.M. (1963), “Information for Estimating the Proportions in Mixtures of Exponential and Normal Distributions”, Journal of the American Statistical Association, 58, 918–932.CrossRefMathSciNetGoogle Scholar
  9. HOLZMANN, H., MUNK, A., and GNEITING, T. (2006), “Identifiability of Finite Mixtures of Elliptical Distributions”, Scandinavian Journal of Statistics, 33, 753–763.zbMATHCrossRefMathSciNetGoogle Scholar
  10. HOLZMANN, H., MUNK, A., and STRATMANN, B. (2004), “Identifiability of Finite Mixtures – with Applications to Circular Distributions”, Sankhyâ, 66, 440–449.MathSciNetGoogle Scholar
  11. LI, L.A. and SEDRANSK, N. (1988), “Mixtures of Distributions: A Topological Approach”, The Annals of Statistics, 16, 1623–1634.zbMATHCrossRefMathSciNetGoogle Scholar
  12. LÜXMANN-ELLINGHAUS, U. (1987), “On the Identifiability of Mixtures of Infinitely Divisible Power Series Distributions”, Statistics and Probability Letters, 5, 375–378.zbMATHCrossRefMathSciNetGoogle Scholar
  13. MCLACHLAN, G.J. and PEEL, D. (2000), Finite Mixture Models, New York: Wiley.zbMATHGoogle Scholar
  14. REDNER, R.A. andWALKER, H.F. (1984), “Mixture Densities, Maximum Likelihood and the EM Algorithm”, SIAM Review, 26, 195–239.zbMATHCrossRefMathSciNetGoogle Scholar
  15. TEICHER, H. (1961), “Identifiability of Mixtures”, The Annals of Mathematical Statistics, 32, 244–248.zbMATHCrossRefMathSciNetGoogle Scholar
  16. TEICHER, H. (1963), “Identifiability of Finite Mixtures”, The Annals of Mathematical Statistics, 34, 1265–1269.zbMATHCrossRefMathSciNetGoogle Scholar
  17. TEICHER, H. (1967), “Identifiability of Mixtures of Product Measures”, The Annals of Mathematical Statistics, 38, 1300–1302.zbMATHCrossRefMathSciNetGoogle Scholar
  18. TITTERINGTON, D.M., SMITH, A.F.M., and MAKOV, U.E. (1985), Statistical Analysis of Finite Mixture Distributions, New York: Wiley.zbMATHGoogle Scholar
  19. WEDEL, M. and KAMAKURA, W.A. (2000), Market Segmentation: Conceptual and Methodological Foundations (2nd ed.), Boston: Kluwer.Google Scholar
  20. YAKOWITZ, S.J. and SPRAGINS, J.D. (1968), “On the Identifiability of Finite Mixtures”, The Annals of Mathematical Statistics, 39, 209–214.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.RAND Corporation and University of GroningenSanta MonicaUSA
  2. 2.University College London and University of GroningenSanta MonicaUSA
  3. 3.RAND CorporationSanta MonicaUSA

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