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On min-max majority and deepest points

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  • Nolan, D.

Abstract

The asymptotic properties of a multivariate location estimator are obtained in this paper. The estimator examined is based on the notion of half-space depth, where the depth of a point is the minimum probability content of all half spaces containing the point. The location estimator of interest is the deepest point with respect to the empirical measure on half spaces. For angularly symmetric distributions, this estimator is consistent. For two dimensions, the exact limit distribution is derived, and the extension of the limit distribution results to higher dimensions is discussed.

Suggested Citation

  • Nolan, D., 1999. "On min-max majority and deepest points," Statistics & Probability Letters, Elsevier, vol. 43(4), pages 325-333, July.
    • Handle: RePEc:eee:stapro:v:43:y:1999:i:4:p:325-333
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    References listed on IDEAS

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    1. Nolan, D., 1992. "Asymptotics for multivariate trimming," Stochastic Processes and their Applications, Elsevier, vol. 42(1), pages 157-169, August.
    2. Caplin, Andrew & Nalebuff, Barry, 1991. "Aggregation and Imperfect Competition: On the Existence of Equilibrium," Econometrica, Econometric Society, vol. 59(1), pages 25-59, January.
    3. Caplin, Andrew S & Nalebuff, Barry J, 1988. "On 64%-Majority Rule," Econometrica, Econometric Society, vol. 56(4), pages 787-814, July.
    4. Caplin, Andrew & Nalebuff, Barry, 1991. "Aggregation and Social Choice: A Mean Voter Theorem," Econometrica, Econometric Society, vol. 59(1), pages 1-23, January.
    5. D. M. Titterington, 1978. "Estimation of Correlation Coefficients by Ellipsoidal Trimming," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 27(3), pages 227-234, November.
    6. Oja, Hannu, 1983. "Descriptive statistics for multivariate distributions," Statistics & Probability Letters, Elsevier, vol. 1(6), pages 327-332, October.
    7. Rousseeuw, Peter J., 1993. "A resampling design for computing high-breakdown regression," Statistics & Probability Letters, Elsevier, vol. 18(2), pages 125-128, September.
    8. Peter J. Rousseeuw & Ida Ruts, 1996. "Bivariate Location Depth," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 45(4), pages 516-526, December.
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    Cited by:

    1. Romanazzi, Mario, 2001. "Influence Function of Halfspace Depth," Journal of Multivariate Analysis, Elsevier, vol. 77(1), pages 138-161, April.
    2. Massé, Jean-Claude, 2002. "Asymptotics for the Tukey Median," Journal of Multivariate Analysis, Elsevier, vol. 81(2), pages 286-300, May.
    3. Kuelbs, James & Zinn, Joel, 2016. "Convergence of quantile and depth regions," Stochastic Processes and their Applications, Elsevier, vol. 126(12), pages 3681-3700.
    4. Eliana Christou, 2020. "Robust dimension reduction using sliced inverse median regression," Statistical Papers, Springer, vol. 61(5), pages 1799-1818, October.

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