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Indistinguishability of absolutely continuous and singular distributions

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  • Lalley, Steven P.
  • Nobel, Andrew

Abstract

It is shown that there are no consistent decision rules for the hypothesis testing problem of distinguishing between absolutely continuous and purely singular probability distributions on the real line. In fact, there are no consistent decision rules for distinguishing between absolutely continuous distributions and distributions supported by Borel sets of Hausdorff dimension 0. It follows that there is no consistent sequence of estimators of the Hausdorff dimension of a probability distribution.

Suggested Citation

  • Lalley, Steven P. & Nobel, Andrew, 2003. "Indistinguishability of absolutely continuous and singular distributions," Statistics & Probability Letters, Elsevier, vol. 62(2), pages 145-154, April.
    • Handle: RePEc:eee:stapro:v:62:y:2003:i:2:p:145-154
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    Cited by:

    1. Marcia M Schafgans & Victoria Zinde-Walshyz, 2008. "Smoothness Adaptive AverageDerivative Estimation," STICERD - Econometrics Paper Series 529, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    2. Yulia Kotlyarova & Marcia M. A. Schafgans & Victoria Zinde-Walsh, 2021. "Rates of Expansions for Functional Estimators," Journal of Quantitative Economics, Springer;The Indian Econometric Society (TIES), vol. 19(1), pages 121-139, December.
    3. Victoria Zinde-Walsh, 2008. "Consequences of lack of smoothness in nonparametric estimation (in Russian)," Quantile, Quantile, issue 4, pages 57-69, March.

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