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Minimum Divergence Estimators Based on Grouped Data

  • M. Menéndez
  • D. Morales
  • L. Pardo
  • I. Vajda
Article

Abstract

The paper considers statistical models with real-valued observations i.i.d. by F(x, θ0) from a family of distribution functions (F(x, θ); θ ε Θ), Θ ⊂ R s , s ≥ 1. For random quantizations defined by sample quantiles (F n −11),θ, F n −1m−1)) of arbitrary fixed orders 0 < λ1 θ < λm-1 < 1, there are studied estimators θφ,n of θ0 which minimize φ-divergences of the theoretical and empirical probabilities. Under an appropriate regularity, all these estimators are shown to be as efficient (first order, in the sense of Rao) as the MLE in the model quantified nonrandomly by (F−110),θ, F−1m−1, θ0)). Moreover, the Fisher information matrix I m 0, λ) of the latter model with the equidistant orders λ = (λ j = j/m : 1 ≤ jm − 1) arbitrarily closely approximates the Fisher information J0) of the original model when m is appropriately large. Thus the random binning by a large number of quantiles of equidistant orders leads to appropriate estimates of the above considered type.

Minimum divergence estimators random quantization asymptotic normality efficiency Fisher information optimization 

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References

  1. Birch, M. W. (1964). A new proof of the Pearson-Fisher theorem, Ann. Math. Statist., 35, 817-824.Google Scholar
  2. Bofinger, E. (1973). Goodness-of-fit using sample quantiles, J. Roy. Statist. Soc. Ser. B, 35, 277-284.Google Scholar
  3. Cheng, R. C. H. (1975). A unified approach to choosing optimum quantiles for the ABLE's, J. Amer. Statist. Assoc., 70, 155-159.Google Scholar
  4. Cressie, N. A. C. and Read, R. C. (1984). Multinomial goodness-of-fit tests, J. Roy. Statist. Soc. Ser. B, 46, 440-464.Google Scholar
  5. Devroye, L., Györfi, L. and Lugosi, G. (1996). A Probabilistic Theory of Pattern Recognition, Springer, New York.Google Scholar
  6. Ferentinos, K. and Papaioannou, T. (1979). Loss of information due to groupings, Transactions of the Eighth Prague Conference on Information Theory, Statistical Decision Functions and Random Processes, 87-94, Prague Academia.Google Scholar
  7. Liese, F. and Vajda, I. (1987). Convex Statistical Distances, Teubner, Leipzig.Google Scholar
  8. Lindsay, B. G. (1994). Efficiency versus robustness: The case for minimum Hellinger distance and other methods, Ann. Statist., 22, 1081-1114.Google Scholar
  9. Menéndez, M. L., Morales, D. and Pardo, L. (1997). Maximum entropy principle and statistical inference on condensed ordered data, Statist. Probab. Lett., 34, 85-93.Google Scholar
  10. Menéndez, M. L., Morales, D., Pardo, L. and Vajda, I. (1998). Two approaches to grouping of data and related disparity statistics, Comm. Statist. Theory Methods, 27(3), 609-633.Google Scholar
  11. Morales, D., Pardo, L. and Vajda, I. (1995). Asymptotic divergence of estimates of discrete distributions, J. Statist. Plann. Inference, 48, 347-369.Google Scholar
  12. Nagahata, H. (1985). Optimal spacing for grouped observations from the information view-point, Mathematica Japonica, 30, 277-282.Google Scholar
  13. Neyman, J. (1949). Contribution to the theory of the X 2 test, Proceeding of the First Berkeley Symposium on Mathematical Statistics and Probability, 239-273. Berkeley University Press, Berkeley, California.Google Scholar
  14. Pötzelberger, K. and Felsenstein, K. (1993). On the Fisher information of discretized data, J. Statist. Comput. Simulation, 46, 125-144.Google Scholar
  15. Rao, C. R. (1961). Asymptotic efficiency and limiting information, Proc. Fourth Berkeley Symp. on Math. Statist. Prob., Vol. 1, 531-545, Berkeley University Press, Berkeley, California.Google Scholar
  16. Rao, C. R. (1973) Linear Statistical Inference and Its Applications, 2nd ed., Wiley, New York.Google Scholar
  17. Tsairidis, Ch., Zografos, K. and Ferentinos, T. (1998). Fisher's information matrix and divergence for finite optimal partitions of the sample space, Comm. Statist. Theory Methods., 26(9), 2271-2289.Google Scholar
  18. Vajda, I. (1973). X 2-divergence and generalized Fisher information, Transactions of the Sixth Prague Conference on Information Theory, Statistical Decision Functions and Random Processes, 223-234, Prague Academia.Google Scholar
  19. Vajda, I. (1989). Theory of Statistical Inference and Information, Kluwer, Boston.Google Scholar

Copyright information

© The Institute of Statistical Mathematics 2001

Authors and Affiliations

  • M. Menéndez
    • 1
  • D. Morales
    • 2
  • L. Pardo
    • 3
  • I. Vajda
    • 4
  1. 1.Department of Applied MathematicsTechnical University of MadridMadridSpain
  2. 2.Operations Research CenterMiguel Hernández University of ElcheElcheSpain
  3. 3.Department of Statistics & O. R.Complutense University of MadridMadridSpain
  4. 4.Institute of Information TheoryAcademy of Sciences of the Czech RepublicPragueCzech Republic

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