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A normal hierarchical model and minimum contrast estimation for random intervals

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  • Yan Sun
  • Dan Ralescu

Abstract

Many statistical data are imprecise due to factors such as measurement errors, computation errors, and lack of information. In such cases, data are better represented by intervals rather than by single numbers. Existing methods for analyzing interval-valued data include regressions in the metric space of intervals and symbolic data analysis, the latter being proposed in a more general setting. However, there has been a lack of literature on the parametric modeling and distribution-based inferences for interval-valued data. In an attempt to fill this gap, we extend the concept of normality for random sets by Lyashenko and propose a Normal hierarchical model for random intervals. In addition, we develop a minimum contrast estimator (MCE) for the model parameters, which is both consistent and asymptotically normal. Simulation studies support our theoretical findings and show very promising results. Finally, we successfully apply our model and MCE to a real data set. Copyright The Institute of Statistical Mathematics, Tokyo 2015

Suggested Citation

  • Yan Sun & Dan Ralescu, 2015. "A normal hierarchical model and minimum contrast estimation for random intervals," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(2), pages 313-333, April.
    • Handle: RePEc:spr:aistmt:v:67:y:2015:i:2:p:313-333
    DOI: 10.1007/s10463-014-0453-1
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    References listed on IDEAS

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    1. Billard L. & Diday E., 2003. "From the Statistics of Data to the Statistics of Knowledge: Symbolic Data Analysis," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 470-487, January.
    2. Dietrich Stoyan, 1998. "Random Sets: Models and Statistics," International Statistical Review, International Statistical Institute, vol. 66(1), pages 1-27, April.
    3. Lothar Heinrich, 1993. "Asymptotic properties of minimum contrast estimators for parameters of boolean models," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 40(1), pages 67-94, December.
    4. J. Pfanzagl, 1969. "On the measurability and consistency of minimum contrast estimates," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 14(1), pages 249-272, December.
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    Cited by:

    1. Yan Sun & Guanghua Lian & Zudi Lu & Jennifer Loveland & Isaac Blackhurst, 2020. "Modeling the Variance of Return Intervals Toward Volatility Prediction," Journal of Time Series Analysis, Wiley Blackwell, vol. 41(4), pages 492-519, July.
    2. Sun, Yan, 2017. "Asymptotic tests for interval-valued means," Statistics & Probability Letters, Elsevier, vol. 121(C), pages 70-77.

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