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Notes on consistency of some minimum distance estimators with simulation results

Author

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  • Jitka Hrabáková

    (Czech Technical University in Prague)

  • Václav Kůs

    (Czech Technical University in Prague)

Abstract

We focus on the minimum distance density estimators $${\widehat{f}}_n$$ f ^ n of the true probability density $$f_0$$ f 0 on the real line. The consistency of the order of $$n^{-1/2}$$ n - 1 / 2 in the (expected) L $$_1$$ 1 -norm of Kolmogorov estimator (MKE) is known if the degree of variations of the nonparametric family $$\mathcal {D}$$ D is finite. Using this result for MKE we prove that minimum Lévy and minimum discrepancy distance estimators are consistent of the order of $$n^{-1/2}$$ n - 1 / 2 in the (expected) L $$_1$$ 1 -norm under the same assumptions. Computer simulation for these minimum distance estimators, accompanied by Cramér estimator, is performed and the function $$s(n)=a_0+a_1\sqrt{n}$$ s ( n ) = a 0 + a 1 n is fitted to the L $$_1$$ 1 -errors of $${\widehat{f}}_n$$ f ^ n leading to the proportionality constant $$a_1$$ a 1 determination. Further, (expected) L $$_1$$ 1 -consistency rate of Kolmogorov estimator under generalized assumptions based on asymptotic domination relation is studied. No usual continuity or differentiability conditions are needed.

Suggested Citation

  • Jitka Hrabáková & Václav Kůs, 2017. "Notes on consistency of some minimum distance estimators with simulation results," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 80(2), pages 243-257, February.
  • Handle: RePEc:spr:metrik:v:80:y:2017:i:2:d:10.1007_s00184-016-0601-0
    DOI: 10.1007/s00184-016-0601-0
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    References listed on IDEAS

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    1. L. Györfi & I. Vajda & E. Meulen, 1996. "Minimum kolmogorov distance estimates of parameters and parametrized distributions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 43(1), pages 237-255, December.
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    3. Alison L. Gibbs & Francis Edward Su, 2002. "On Choosing and Bounding Probability Metrics," International Statistical Review, International Statistical Institute, vol. 70(3), pages 419-435, December.
    4. Broniatowski, Michel, 2014. "Minimum divergence estimators, maximum likelihood and exponential families," Statistics & Probability Letters, Elsevier, vol. 93(C), pages 27-33.
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