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Convergence of quantile and depth regions

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  • Kuelbs, James
  • Zinn, Joel

Abstract

Since contours of multi-dimensional depth functions often characterize the distribution, it has become of interest to consider structural properties and limit theorems for the sample contours (see Zuo and Serfling (2000)). For finite dimensional data Massé and Theodorescu (1994) [14] and Kong and Mizera (2012) have made connections of directional quantile envelopes to level sets of half-space (Tukey) depth. In the recent paper (Kuelbs and Zinn, 2014) we showed that half-space depth regions determined by evaluation maps of a stochastic process are not only uniquely determined by related upper and lower quantile functions for the process, but limit theorems have also been obtained. In this paper we study the consequences of these results when applied to finite dimensional data in greater detail. The methods we employ here are based on Kuelbs and Zinn (2015) and Kuelbs and Zinn (2013).

Suggested Citation

  • Kuelbs, James & Zinn, Joel, 2016. "Convergence of quantile and depth regions," Stochastic Processes and their Applications, Elsevier, vol. 126(12), pages 3681-3700.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:12:p:3681-3700
    DOI: 10.1016/j.spa.2016.04.011
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    References listed on IDEAS

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    1. Masse, J. C. & Theodorescu, R., 1994. "Halfplane Trimming for Bivariate Distributions," Journal of Multivariate Analysis, Elsevier, vol. 48(2), pages 188-202, February.
    2. Nolan, D., 1999. "On min-max majority and deepest points," Statistics & Probability Letters, Elsevier, vol. 43(4), pages 325-333, July.
    3. López-Pintado, Sara & Romo, Juan, 2009. "On the Concept of Depth for Functional Data," Journal of the American Statistical Association, American Statistical Association, vol. 104(486), pages 718-734.
    4. López-Pintado, Sara & Romo, Juan, 2011. "A half-region depth for functional data," Computational Statistics & Data Analysis, Elsevier, vol. 55(4), pages 1679-1695, April.
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