Scatter halfspace depth: Geometric insights

Stanislav Nagy

Applications of Mathematics (2020)

  • Volume: 65, Issue: 3, page 287-298
  • ISSN: 0862-7940

Abstract

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Scatter halfspace depth is a statistical tool that allows one to quantify the fitness of a candidate covariance matrix with respect to the scatter structure of a probability distribution. The depth enables simultaneous robust estimation of location and scatter, and nonparametric inference on these. A handful of remarks on the definition and the properties of the scatter halfspace depth are provided. It is argued that the currently used notion of this depth is well suited especially for symmetric random vectors. The scatter halfspace depth closely relates to an appropriate distance of matrix-generated ellipsoids from an upper level set of the (location) halfspace depth function. Several modifications and extensions to the scatter halfspace depth are considered, with their theoretical properties outlined.

How to cite

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Nagy, Stanislav. "Scatter halfspace depth: Geometric insights." Applications of Mathematics 65.3 (2020): 287-298. <http://eudml.org/doc/296935>.

@article{Nagy2020,
abstract = {Scatter halfspace depth is a statistical tool that allows one to quantify the fitness of a candidate covariance matrix with respect to the scatter structure of a probability distribution. The depth enables simultaneous robust estimation of location and scatter, and nonparametric inference on these. A handful of remarks on the definition and the properties of the scatter halfspace depth are provided. It is argued that the currently used notion of this depth is well suited especially for symmetric random vectors. The scatter halfspace depth closely relates to an appropriate distance of matrix-generated ellipsoids from an upper level set of the (location) halfspace depth function. Several modifications and extensions to the scatter halfspace depth are considered, with their theoretical properties outlined.},
author = {Nagy, Stanislav},
journal = {Applications of Mathematics},
keywords = {elliptical distributions; floating body; scatter halfspace depth; Tukey depth},
language = {eng},
number = {3},
pages = {287-298},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Scatter halfspace depth: Geometric insights},
url = {http://eudml.org/doc/296935},
volume = {65},
year = {2020},
}

TY - JOUR
AU - Nagy, Stanislav
TI - Scatter halfspace depth: Geometric insights
JO - Applications of Mathematics
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 3
SP - 287
EP - 298
AB - Scatter halfspace depth is a statistical tool that allows one to quantify the fitness of a candidate covariance matrix with respect to the scatter structure of a probability distribution. The depth enables simultaneous robust estimation of location and scatter, and nonparametric inference on these. A handful of remarks on the definition and the properties of the scatter halfspace depth are provided. It is argued that the currently used notion of this depth is well suited especially for symmetric random vectors. The scatter halfspace depth closely relates to an appropriate distance of matrix-generated ellipsoids from an upper level set of the (location) halfspace depth function. Several modifications and extensions to the scatter halfspace depth are considered, with their theoretical properties outlined.
LA - eng
KW - elliptical distributions; floating body; scatter halfspace depth; Tukey depth
UR - http://eudml.org/doc/296935
ER -

References

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  1. Chen, M., Gao, C., Ren, Z., 10.1214/17-AOS1607, Ann. Stat. 46 (2018), 1932-1960. (2018) Zbl1408.62104MR3845006DOI10.1214/17-AOS1607
  2. Dupin, C., Applications de géométrie et de méchanique à la marine, aux ponts et chaussées, etc. pour faire suite aux développements de géométrie, Bachelier, Paris (1822), French. (1822) 
  3. Fang, K.-T., Kotz, S., Ng, K.-W., 10.1201/9781351077040, Monographs on Statistics and Applied Probability 36, Chapman and Hall, London (1990). (1990) Zbl0699.62048MR1071174DOI10.1201/9781351077040
  4. Helgason, S., 10.1007/978-1-4419-6055-9, Springer, New York (2011). (2011) Zbl1210.53002MR2743116DOI10.1007/978-1-4419-6055-9
  5. Meyer, M., Schütt, C., Werner, E. M., 10.1007/s11856-015-1196-2, Isr. J. Math. 208 (2015), 163-192. (2015) Zbl1343.52003MR3416917DOI10.1007/s11856-015-1196-2
  6. Nagy, S., 10.1016/j.spl.2019.02.006, Stat. Probab. Lett. 149 (2019), 171-177. (2019) Zbl1427.62041MR3921058DOI10.1016/j.spl.2019.02.006
  7. Nagy, S., The halfspace depth characterization problem, (to appear) in Springer Proc. Math. Stat. (2020). 
  8. Nagy, S., Schütt, C., Werner, E. M., 10.1214/19-ss123, Stat. Surv. 13 (2019), 52-118. (2019) Zbl1428.62204MR3973130DOI10.1214/19-ss123
  9. Paindaveine, D., Bever, G. Van, 10.1214/17-AOS1658, Ann. Stat. 46 (2018), 3276-3307. (2018) Zbl1408.62100MR3852652DOI10.1214/17-AOS1658
  10. Quinto, E. T., 10.1016/0022-247X(83)90118-X, J. Math. Anal. Appl. 95 (1983), 437-448. (1983) Zbl0569.44005MR716094DOI10.1016/0022-247X(83)90118-X
  11. Schneider, R., 10.1017/CBO9781139003858, Encyclopedia of Mathematics and Its Applications 151, Cambridge University Press, Cambridge (2014). (2014) Zbl1287.52001MR3155183DOI10.1017/CBO9781139003858
  12. Tukey, J. W., Mathematics and the picturing of data, Proceedings of the International Congress of Mathematicians. Vol. 2 Canadian Mathematical Society, Vancouver (1974), 523-531. (1974) Zbl0347.62002MR0426989

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