Electronic Journal of Statistics

A new concept of quantiles for directional data and the angular Mahalanobis depth

Christophe Ley, Camille Sabbah, and Thomas Verdebout

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In this paper, we introduce a new concept of quantiles and depth for directional (circular and spherical) data. In view of the similarities with the classical Mahalanobis depth for multivariate data, we call it the angular Mahalanobis depth. Our unique concept combines the advantages of both the depth and quantile settings: appealing depth-based geometric properties of the contours (convexity, nestedness, rotation-equivariance) and typical quantile-asymptotics, namely we establish a Bahadur-type representation and asymptotic normality (these results are corroborated by a Monte Carlo simulation study). We introduce new user-friendly statistical tools such as directional DD- and QQ-plots and a quantile-based goodness-of-fit test. We illustrate the power of our new procedures by analyzing a cosmic rays data set.

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Electron. J. Statist., Volume 8, Number 1 (2014), 795-816.

First available in Project Euclid: 16 June 2014

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Bahadur representation directional statistics DD- and QQ-plot Mahalanobis depth rotationally symmetric distributions


Ley, Christophe; Sabbah, Camille; Verdebout, Thomas. A new concept of quantiles for directional data and the angular Mahalanobis depth. Electron. J. Statist. 8 (2014), no. 1, 795--816. doi:10.1214/14-EJS904. https://projecteuclid.org/euclid.ejs/1402927498

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  • [1] Agostinelli, C. and Romanazzi, M. (2013). Nonparametric analysis of directional data based on data depth. Environ. Ecol. Stat., 20, 253–270.
  • [2] Bahadur, R. R. (1966). A note on quantiles in large samples. Ann. Math. Statist., 37, 577–580.
  • [3] Boomsma, W., Kent, J. T., Mardia, K. V., Taylor, C. C. and Hamelryck, T. (2006). Graphical models and directional statistics capture protein structure. In S. Barber, P. D. Baxter, K. V. Mardia & R. E. Walls (Eds.), LASR 2006–Interdisciplinary Statistics and Bioinformatics, Leeds University Press, UK, 91–94.
  • [4] Bowley, A. L. (1902). Elements of Statistics, 2nd edition. P. S. King, London.
  • [5] Chaudhuri, P. (1996). On a geometric notion of quantiles for multivariate data. J. Amer. Statist. Assoc., 91, 862–872.
  • [6] Efron, B. (1979). Bootstrap methods: Another look at the jackknife. Ann. Statist., 7, 1–26.
  • [7] Fisher, N. I. (1985). Spherical medians. J. Roy. Stat. Soc. B, 47, 342–348.
  • [8] Hallin, M., Paindaveine, D. and Siman, M. (2010). Multivariate quantiles and multiple-output regression quantiles: from $L_{1}$ optimization to halfspace depth. Ann. Statist., 38, 635–669.
  • [9] Hjort, N. and Pollard, D. (1993). Asymptotics for minimisers of convex processes. Unpublished manuscript, http://www.stat.yale.edu/~pollard/Papers/.
  • [10] Koenker, R. (2005). Quantile Regression, 1st edition. Cambridge University Press, New York.
  • [11] Koenker, R. and Bassett, G. J. (1978). Regression quantiles. Econometrica, 46, 33–50.
  • [12] Kong, L. and Mizera, I. (2012). Quantile tomography: Using quantiles with multivariate data. Statist. Sinica, 22, 1589–1610.
  • [13] Koshevoy, G. and Mosler, K. (1997). Zonoid trimming for multivariate distributions. Ann. Statist., 25, 1998–2017.
  • [14] Kreiss, J. P. (1987). On adaptive estimation in stationary ARMA processes. Ann. Statist., 15, 112–133.
  • [15] LaRiccia, V. N. (1991). Smooth goodness-of-fit tests: A quantile function approach. J. Amer. Statist. Assoc., 86, 427–431.
  • [16] Le Cam, L. and Yang, G. L. (2000). Asymptotics in Statistics, 2nd edition. Springer-Verlag, New York.
  • [17] Lewis, T. and Fisher, N. I. (1982). Graphical methods for investigating the fit of a Fisher distribution to spherical data. Geophys. J. R. astr. Soc., 69, 1–13.
  • [18] Ley, C., Swan, Y., Thiam, B. and Verdebout, T. (2013). Optimal R-estimation of a spherical location. Statist. Sinica, 23, 305–332.
  • [19] Li, J., Cuesta-Albertos, J. and Liu, R. Y. (2012). Dd-classifier: Nonparametric classification procedures based on dd-plots. J. Amer. Statist. Assoc., 107, 737–753.
  • [20] Liu, R. Y. (1990). On a notion of data depth based on random simplices. Ann. Statist., 18, 405–414.
  • [21] Liu, R. Y. (1992). Data depth and multivariate rank tests. In Y. Dodge (Ed.), L-1 Statistics and Related Methods, North-Holland, Amsterdam, 279–294.
  • [22] Liu, R. Y., Parelius, J. M. and Singh, K. (1999). Multivariate analysis by data depth: Descriptive statistics, graphics and inference (with discussion). Ann. Statist., 27, 783–858.
  • [23] Liu, R. Y., Serfling, R. J. and Souvaine, D. L. (Eds.) (2006). Data Depth: Robust Multivariate Analysis, Computational Geometry and Applications. Amer. Math. Soc.
  • [24] Liu, R. Y. and Singh, K. (1992). Ordering directional data: Concept of data depth on circles and spheres. Ann. Statist., 20, 1468–1484.
  • [25] Mardia, K. V. and Jupp, P. E. (2000). Directional Statistics. Wiley, Chichester.
  • [26] Moors, J. J. A. (1988). A quantile alternative for kurtosis. J. Roy. Stat. Soc. D, 37, 25–32.
  • [27] Mosler, K. (2013). Depth statistics. In C. Becker, R. Fried. S. Kuhnt (Eds.), Robustness and Complex Data Structures, Festschrift in Honor of Ursula Gather, Berlin, Springer, 17–34.
  • [28] Purkayastha, S. (1991). A rotationally symmetric directional distribution: Obtained through maximum likelihood characterization. Sankhya Ser. A, 53, 70–83.
  • [29] Rousseeuw, P. J. and Hubert, M. (1999). Regression depth (with discussion). J. Amer. Statist. Assoc., 94, 388–433.
  • [30] Serfling, R. (2002). Quantile functions for multivariate analysis: Approaches and applications. Statist. Neerlandica, 56, 214–232. Special issue: Frontier research in theoretical statistics, 2000 (Eindhoven).
  • [31] Small, C. G. (1990). A survey of multidimensional medians. International Statistical Review, 58, 263–277.
  • [32] Toyoda, Y., Suga, K., Murakami, K., Hasegawa, H., Shibata, S., Domingo, V., Escobar, I., Kamata, K., Bradt, H., Clark, G. and La Pointe, M. (1965). Studies of primary cosmic rays in the energy region $10^{14}$ eV to $10^{17}$ eV (Bolivian Air Shower Joint Experiment). Proc. Int. Conf. Cosmic Rays (London), 2, 708–711.
  • [33] Tukey, J. W. (1975). Mathematics and the picturing of data. In Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 2, Canad. Math. Congress, Montreal, Quebec, 523–531.
  • [34] Watson, G. S. (1983). Statistics on Spheres. Wiley, New York.
  • [35] Zuo, Y. and Serfling, R. (2000). General notions of statistical depth function. Ann. Statist., 28, 461–482.