Minimum Divergence Estimators Based on Grouped Data

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


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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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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