Skip to main content
SpringerLink
Log in

Mixture Models and Nonparametric Deconvolution

  • Chapter
Asymptotic Theory of Statistics and Probability

Part of the book series: Springer Texts in Statistics ((STS))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 79.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Barndorff-Nielsen, O. (1978). Hyperbolic distributions and distributions on hyperbolae, Scand. J. Stat., 5, 151–157.

    MathSciNet  MATH  Google Scholar 

  • Barndorff-Nielsen, O., Kent, J., and Sorensen, M. (1982). Normal variance-mean mixtures and z distributions, Int. Stat. Rev., 50, 145–159.

    Article  MATH  MathSciNet  Google Scholar 

  • Basu, A., Harris, I., and Basu, S. (1997). Minimum distance estimation: the approach using density based divergences, in Handbook of Statistics, Vol. 15, Maddala, G. and Rao, C.R. (eds.), North-Holland, Amsterdam, 21–48.

    Google Scholar 

  • Behboodian, J. (1970). On a mixture of normal distributions, Biometrika, 57 (1), 215–217.

    Article  MATH  Google Scholar 

  • Beran, R. (1977). Minimum Hellinger distance estimates for parametric models, Ann. Stat., 5(3), 445–463.

    Article  MATH  MathSciNet  Google Scholar 

  • Blischke, W. (1962). Moment estimators for parameters of mixtures of two Binomial distributions, Ann. Math. Stat., 33, 444–454.

    Article  MathSciNet  MATH  Google Scholar 

  • Bowman, K. and Shenton, L. (1973). Space of solutions for a normal mixture, Biometrika, 60, 629–636.

    Article  MATH  MathSciNet  Google Scholar 

  • Carroll, R. and Hall, P. (1988). Optimal rates of convergence for deconvolving a density, J. Am. Stat. Assoc., 83(404), 1184–1186.

    Article  MATH  MathSciNet  Google Scholar 

  • Champeney, D. (2003). Handbook of Fourier Theorems, Cambridge University Press, Cambridge.

    Google Scholar 

  • Chen, J. (1995). Optimal rate of convergence for finite mixture models, Ann. Stat., 23(1), 221–233.

    Article  MATH  Google Scholar 

  • Chen, J. and Kalbfleisch, J. (1996). Penalized minimum distance estimates in finite mixture models, Can. J. Stat., 24 (2), 167–175.

    Article  MATH  MathSciNet  Google Scholar 

  • Cheney, W. and Light, L. (1999). A Course in Approximation Theory, Brooks and Cole, Boston CA.

    Google Scholar 

  • Choi, K. and Bulgren, W. (1967). An estimation procedure for mixtures of distributions, J.R. Stat. Soc. B, 30, 444–460.

    MathSciNet  Google Scholar 

  • Cohen, A.C. (1967). Estimation in mixtures of two normal distributions, Technometrics, 9(1), 15–28.

    Article  MATH  MathSciNet  Google Scholar 

  • Cutler, A. and Cordero-Brana, O. (1996). Minimum Hellinger distance estimation for finite mixture models, J. Am. Stat. Assoc., 91 (436), 1716–1723.

    Article  MATH  MathSciNet  Google Scholar 

  • Dempster, A., Laird, N., and Rubin, D. (1977). Maximum likelihood from incomplete data via the EM algorithm, with discussion, J. R. Stat. Soc. B, 39 (1), 1–38.

    MATH  MathSciNet  Google Scholar 

  • Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems, Ann. Stat., 19 (3), 1257–1272.

    Article  MATH  Google Scholar 

  • Feller, W. (1966). An Introduction to Probability Theory and its Applications, Vol. II, Wiley, New York.

    MATH  Google Scholar 

  • Fisher, R.A. (1921). On the probable error of a coefficient of correlation deduced from a small sample, Metron, 1, 3–32.

    Google Scholar 

  • Ghosal, S. and van der Vaart, A. (2001). Entropies and rates of convergence for maximum likelihood and Bayes estimation for mixtures of normal densities, Ann. Stat., 29 (5), 1233–1263.

    Article  MATH  Google Scholar 

  • Ghosh, J. and Sen, P.K. (1985). On the asymptotic performance of the log-likelihood ratio statistic for the mixture model, in Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, L. Le Cam and R.Olshen (eds.), Wadsworth, Belmont, CA, 789–806.

    Google Scholar 

  • Hall, P. and Stewart, M. (2005). Theoretical analysis of power in a two component normal mixture model, J. Stat. Planning Infer., 134 (1), 158–179.

    Article  MATH  MathSciNet  Google Scholar 

  • Hall, P. and Titterington, D. (1984). Efficient nonparametric estimation of mixture proportions, J. R. Stat. Soc. B., 46 (3), 465–473.

    MATH  MathSciNet  Google Scholar 

  • Hall, P. and Zhou, X. (2003). Nonparametric estimation of component distributions in a multivariate mixture, Ann. Stat., 31 (1), 201–224.

    Article  MATH  MathSciNet  Google Scholar 

  • Hartigan, J. (1985). A failure of likelihood asymptotics for normal mixtures, in Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, L. Le Cam and R.Olshen (eds.), Wadsworth, Belmont, CA, 807–810.

    Google Scholar 

  • Hasselblad, V. (1966). Estimation of parameters for a mixture of normal distributions, Technometrics, 8, 431–444.

    Article  MathSciNet  Google Scholar 

  • Hosmer, D. (1973). On MLE of the parameters of the mixture of two normal distributions when the sample size is small, Commun. Stat., 1, 217–227.

    Article  MathSciNet  Google Scholar 

  • Jewell, N. (1982). Mixtures of exponential distributions, Ann. Stat., 10, 479–484.

    Article  MATH  MathSciNet  Google Scholar 

  • John, S. (1970). On identifying the population of origin of each observation in a mixture of observations from two gamma populations, Technometrics, 12, 565–568.

    Article  Google Scholar 

  • Johnson, N. and Kotz, S. (1969). Distributions in Statistics: Continuous Univariate Distributions, Vol. 2, Houghton Mifflin, Boston.

    Google Scholar 

  • Kabir, A. (1968). Estimation of parameters of a finite mixture of distributions, J. R. Stat. Soc. B, 30, 472–482.

    Google Scholar 

  • Kiefer, J. and Wolfowitz, J. (1956). Consistency of the maximum likelihood estimator in the presence of infinitely many nuisance parameters, Ann. Math. Stat., 27, 887–906.

    Article  MathSciNet  MATH  Google Scholar 

  • Laird, N. (1978). Nonparametric maximum likelihood estimation of a mixing distribution, J. Am. Stat. Assoc., 73, 805–811.

    Article  MATH  MathSciNet  Google Scholar 

  • Lambert, D. and Tierney, L. (1984). Asymptotic properties of maximum likelihood estimates in the mixed Poisson model, Ann. Stat., 12(4), 1388–1399.

    Article  MATH  MathSciNet  Google Scholar 

  • Lindsay, B. (1983a). The geometry of mixture likelihoods: a general theory, Ann. Stat., 11 (1), 86–94.

    Article  MATH  MathSciNet  Google Scholar 

  • Lindsay, B. (1983b). The geometry of mixture likelihoods II: the Exponential family, Ann. Stat., 11(3), 783–792.

    Article  MATH  MathSciNet  Google Scholar 

  • Lindsay, B. (1995). Mixture Models: Theory, Geometry, and Applications, NSF-CBMS Series in Probability and Statistics, Institute of Mathematical Statistics, Hayward, CA.

    Google Scholar 

  • Lindsay, B. and Basak, P. (1993). Multivariate normal mixtures: a fast consistent method of moments, J. Am. Stat. Assoc., 88 (422), 468–476.

    Article  MATH  MathSciNet  Google Scholar 

  • McLachlan, G. and Peel, D. (2000). Finite Mixture Models, John Wiley, New York.

    MATH  Google Scholar 

  • Pearson, K. (1894). Contributions to the mathematical theory of evolution, Philos. Trans. R. Soc. A, 185, 71–110.

    Article  Google Scholar 

  • Peters, B. and Walker, H. (1978). An iterative procedure for obtaining maximum likelihood estimates of the parameters for a mixture of normal distributions, SIAM J. Appl. Math., 35, 362–378.

    Article  MATH  MathSciNet  Google Scholar 

  • Pfanzagl, J. (1988). Consistency of maximum likelihood estimators for certain nonparametric families, in particular mixtures, J. Stat. Planning Infer., 19(2), 137–158.

    Google Scholar 

  • Prentice, R. (1975). Discrimination among some parametric models, Biometrika, 62, 607–614.

    Article  MATH  MathSciNet  Google Scholar 

  • Redner, R. (1981). Consistency of the maximum likelihood estimate for nonidentifiable distributions, Ann. Stat., 9(1), 225–228.

    Google Scholar 

  • Redner, R. and Walker, H. (1984). Mixture densities, maximum likelihood, and the EM algorithm, SIAM Rev., 26 (2), 195–240.

    Article  MATH  MathSciNet  Google Scholar 

  • Rider, P. (1962). Estimating the parameters of mixed Poisson, binomial, and Weibull distributions by the method of moments, Bull. Int. Stat. Inst., 39(2), 225–232.

    MATH  Google Scholar 

  • Roeder, K. (1992). Semiparametric estimation of normal mixture densities, Ann. Stat., 20(2), 929–943.

    Google Scholar 

  • Roeder, K. and Wasserman, L. (1997). Practical Bayesian density estimation using mixtures of normals, J. Am. Stat. Assoc., 92 (439), 894–902.

    Article  MATH  MathSciNet  Google Scholar 

  • Simar, L. (1976). Maximum likelihood estimation of a compound Poisson process, Ann. Stat., 4, 1200–1209.

    Google Scholar 

  • Stefanski, L. and Carroll, R. (1990). Deconvoluting kernel density estimators, Statistics, 21 (2), 169–184.

    Article  MATH  MathSciNet  Google Scholar 

  • Tamura, R. and Boos, D. (1986). Minimum Hellinger distance estimation for multivariate location and covariance, J. Am. Stat. Assoc., 81 (393), 223–229.

    Article  MATH  MathSciNet  Google Scholar 

  • Teicher, H. (1961). Identifiability of mixtures, Ann. Math. Stat., 32, 244–248.

    Google Scholar 

  • Titterington, D., Smith, A., and Makov, U. (1985). Statistical Analysis of Finite Mixture Distributions, John Wiley, New York.

    Google Scholar 

  • van de Geer, S. (1996). Rates of convergence for the maximum likelihood estimator in mixture models, J. Nonparamet. Stat., 6, 293–310.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, Purdue University, 150 North University Street, West Lafayette, IN 47907

    Anirban DasGupta

Authors
  1. Anirban DasGupta
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

DasGupta, A. (2008). Mixture Models and Nonparametric Deconvolution. In: Asymptotic Theory of Statistics and Probability. Springer Texts in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-0-387-75971-5_33

Download citation

Publish with us

Policies and ethics

Search

Navigation