Estimating the Mixing Density of a Mixture of Power Series Distributions

  • Wei-Liem Loh


Let X 1, X 2, ..., X n be independent and identically distributed observations from a mixture of power series distributions. Based on this random sample, we consider the problem of estimating the mixing density of the mixture distribution. A mixing density kernel estimator is proposed which, under mild assumptions, has 1/log n as an upper bound for its rate of convergence to the true density under squared error loss. It is also shown that the optimal rate of convergence cannot exceed 1/n r for any constant r.


Optimal Rate Mixture Distribution Error Loss Deconvolution Problem Compound Poisson Distribution 
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  1. Blum, J. R. and Susarla, V. (1977). Estimation of a mixing distribution function. Ann. Probab. 5 200–209.MathSciNetMATHCrossRefGoogle Scholar
  2. Carroll, R. J. and Hall, P. (1988). Optimal rates of convergence for deconvolving a density. J. Amer. Statist. Assoc. 83 1184–1186.MathSciNetMATHCrossRefGoogle Scholar
  3. Datta, S. (1991). On the consistency of posterior mixtures and its applications. Ann. Statist. 19 338–353.MathSciNetMATHCrossRefGoogle Scholar
  4. Deely, J. J. and Kruse, R. L. (1968). Construction of sequences estimating the mixing distribution. Ann. Math. Statist. 39 286–288.MathSciNetMATHCrossRefGoogle Scholar
  5. Devroye, L. P. (1989). Consistent deconvolution in density estimation. Canad. J. Statist. 17 235–239.MathSciNetMATHCrossRefGoogle Scholar
  6. Devroye, L. P. and Wise, G. L. (1979). On the recovery of discrete probability densities from imperfect measurements. J. Franklin Inst. 307 1–20.MathSciNetMATHCrossRefGoogle Scholar
  7. Donoho, D. L. and Liu, R. C. (1987). Geometrizing rates of convergence, I. Technical Report, Dept. Statist., Univ. California, Berkeley.Google Scholar
  8. Donoho, D. L. and Liu, R. C. (1991a). Geometrizing rates of convergence, II. Ann. Statist. 19 633–667.MathSciNetMATHCrossRefGoogle Scholar
  9. Donoho, D. L. and Liu, R. C. (1991b). Geometrizing rates of convergence, III. Ann. Statist. 19 668–701.MathSciNetMATHCrossRefGoogle Scholar
  10. Fan, J. (1991a). On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 1257–1272.MathSciNetMATHCrossRefGoogle Scholar
  11. Fan, J. (1991b). Global behavior of deconvolution kernel estimates. Statist. Sinica 1 541–551.MathSciNetMATHGoogle Scholar
  12. Fan, J. (1991c). Adaptively local 1-dimensional subproblems. Preprint.Google Scholar
  13. Harris, I. R. (1991). The estimated frequency of zero for a mixed Poisson distribution. Statist. Probab. Lett. 12 371–372.MathSciNetMATHCrossRefGoogle Scholar
  14. Jewell, N. (1982). Mixtures of exponential distributions. Ann. Statist. 10 479–484.MathSciNetMATHCrossRefGoogle Scholar
  15. Johnson, N. L. and Kotz, S. (1969). Discrete Distributions. Wiley, New York.MATHGoogle Scholar
  16. Lambert, D. and Tierney, L. (1984). Asymptotic properties of maximum likelihood estimates in the mixed Poisson model. Ann. Statist. 12 1388–1399.MathSciNetMATHCrossRefGoogle Scholar
  17. Lindsay, B. G. (1983a). The geometry of mixture likelihoods: A general theory. Ann. Statist. 11 86–94.MathSciNetMATHCrossRefGoogle Scholar
  18. Lindsay, B. G. (1983b). The geometry of mixture likelihoods, part II: The exponential family. Ann. Statist. 11 783–792.MathSciNetMATHCrossRefGoogle Scholar
  19. Lindsay, B. G. (1989). Moment matrices: Applications in mixtures. Ann. Statist. 17 722–740.MathSciNetMATHCrossRefGoogle Scholar
  20. Meeden, G. (1972). Bayes estimation of the mixing distribution, the discrete case. Ann. Math. Statist. 43 1993–1999.MathSciNetMATHCrossRefGoogle Scholar
  21. Robbins, H. (1964). The empirical Bayes approach to statistical decision problems. Ann. Math. Statist. 35 1–20.MathSciNetMATHCrossRefGoogle Scholar
  22. Rolph, J. E. (1968). Bayesian estimation of mixing distributions. Ann. Math. Statist. 39 1289–1302.MathSciNetMATHCrossRefGoogle Scholar
  23. Shohat, J. and Tamarkin, J. (1943). The Problem of Moments. Amer. Math. Soc. Waverly Press, Baltimore.Google Scholar
  24. Simar, L. (1976). Maximum likelihood estimation of a compound Poisson process. Ann. Statist. 4 1200–1209.MathSciNetMATHCrossRefGoogle Scholar
  25. Stefanski, L. A. (1990). Rates of convergence of some estimators in a class of deconvolution problems. Statist. Probab. Lett. 9 229–235.MathSciNetMATHCrossRefGoogle Scholar
  26. Stefanski, L. A. and Carroll, R. J. (1990). Deconvolving kernel density estimators. Statist. 21 169–184.MathSciNetMATHGoogle Scholar
  27. Tucker, H. G. (1963). An estimate of the compounding distribution of a compound Poisson distribution. Theor. Probab. Appl. 8 195–200.CrossRefGoogle Scholar
  28. Wise, G. L., Traganitis, A. P. and Thomas, J. B. (1977). The estimation of a probability density function from measurements corrupted by Poisson noise. IEEE Trans. Inform. Theory 23 764–766.MathSciNetMATHCrossRefGoogle Scholar
  29. Zhang, C. H. (1990). Fourier methods for estimating mixing densities and distributions. Ann. Statist. 18 806–831.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1994

Authors and Affiliations

  • Wei-Liem Loh
    • 1
  1. 1.Purdue UniversityUSA

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