Abstract
Depth functions have many applications in multivariate data analysis, including discriminant analysis and classification. In this paper, we introduce a novel class of data depth: exponential power depth (EPD) functions. Under some conditions, we show that the EPD functions are a statistical depth function, and the sample EPD functions are consistent and asymptotically normal. Based on the proposed EPD functions, we construct a DD-plot (depth-versus-depth plot), which can be applied to the classification problem. Since the EPD functions contain the two tuning parameters, we provide a data-driven approach to select these tuning parameters. The simulation studies and two real data analysis are conducted to assess the finite sample performance of the proposed method.
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BARNETT, V. (1976), “The Ordering of Multivariate Data”, Journal of the Royal Statistical Society Series A, 139(3), 318–355.
BUTLER, R., DAVIES, P., and JHUN, M. (1993), “Asymptotics for the Minimum Covariance Determinant Estimator”, Annals of Statististics, 21(3), 1385–1400.
CATOR, E., and LOPUHAA, H. (2010), “Asymptotic Expansion of the Minimum Covariance Determinant Estimators”, Journal of Multivariate Analysis, 101(10), 2372–2388.
CHAKRABORTY, B., CHAUDHURI, P., and OJA, H. (1998), “Operating Transformation Retransformation on Spatial Median and Angle Test”, Statistica Sinica, 8, 767–784.
CHEN, Y., DANG, X., PENG, H., and BART, H. (2009), “Outlier Detection with the Kernelized Spatial Depth Function”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 31(2), 288–305.
CROUX, C., and HAESBROECK, G. (1999), “Influence Function and Efficiency of the Minimum Covariance Determinant Scatter Matrix Estimator”, Journal of Multivariate Analysis, 71(2), 161–190.
CUEVAS, A., FEBRERO, M., and FRAIMAN, R. (2007), “Robust Estimation and Classification for Functional Data Via Projection-Based Depth Notions”, Computational Statistics, 22(3), 481–496.
DAVIES, P.L. (1987), “Asymptotic Behavior of S-Estimators of Multivariate Location Parameters and Dispersion Matrices”, Annals of Statistics, 15(3), 1269–1292.
DONOHO, D. (1982), “Breakdown Properties of Multivariate Location Estimators”, unpublished Ph.D. qualifying paper, Department of Statistics, Harvard University, Boston.
DUTTA, S., and GHOSH, A. (2011), “On Classification Based on l p-Depth with an Adaptive Choice of p”, Technical Report Number R5/2011, Statistics and Mathematics Unit, Indian Statistical Institute, Kolkata, India.
DUTTA, S., and GHOSH, A. (2012), “On Robust Classification Using Projection Depth”, Annals of the Institute of Statistical Mathematics, 64(3), 657–676.
FRAIMAN, R., MELOCHE, J., GARCÍA-ESCUDERO, L., GORDALIZA, A., HE, X., MARONNA, R., YOHAI, V., SHEATHER, S., MCKEAN, J., SMALL, C. et al. (1999), “Multivariate l-Estimation”, Test, 8(2), 255–317.
GHOSH, A., and CHAUDHURI, P. (2005), “On Data Depth and Distribution-Free Discriminant Analysis Using Separating Surfaces”, Bernoulli, 11(1), 1–27.
JIANG, Y.,WANG, S., GE, W., and WANG, X. (2011), “Depth-Based Weighted Empirical Likelihood and General Estimating Equations”, Journal of Nonparametric Statistics, 23(4), 1051–1062.
LI, J., CUESTA-ALBERTOS, J.A., and LIU, R.Y. (2012), “DD-Classifier: Nonparametric Classification Procedure Based on DD-Plot”, Journal of the American Statistical Association, 107(498), 737–753.
LI, J., and LIU, R. (2008), “Multivariate Spacings Based on Data Depth: I. Construction of Nonparametric Multivariate Tolerance Regions”, Annals of Statistics, 36(3), 1299–1323.
LIU, R. (1990), “On a Notion of Data Depth Based Upon Random Simplices”, Annals of Statistics, 18(1), 405–414.
LIU, R. (1995), “Control Charts for Multivariate Processes”, Journal of the American Statistical Association, 90(432), 1380–1387.
LIU, X., and ZUO, Y. (2012), “Computing Projection Depth and Its Associated Estimators”, Statistics and Computing, 1–13.
LÓPEZ-PINTADO, S., and ROMO, J. (2006), “Depth-Based Classification for Functional Data”, DIMACS: Series in Discrete Mathematics and Theoretical Computer Science, 72, 17–35.
MAHALANOBIS, P. (1936), “On the Generalized Distance in Statistics”, Proceedings of the National Academy of Sciences India, 2(1), 49–55.
MARONNA, R. (1976), “Robust m-Estimators of Multivariate Location and Scatter”, Annals of Statistics, 4(1), 51–67.
ROUSSEEUW, P. (1985), “Multivariate Estimation with High Breakdown Point”, Mathematical Statistics and Applications, 8, 283–297.
ROUSSEEUW, P., and DRIESSEN, K. (1999), “A Fast Algorithm for the Minimum Covariance Determinant Estimator”, Technometrics, 41(3), 212–223.
ROUSSEEUW, P., VAN AELST, S., and HUBERT, M. (1999), “Regression Depth: Rejoinder”, Journal of the American Statistical Association, 94(446), 419–433.
SÁNCHEZ-MANZANO, E., GOMEZ-VILLEGAS, M., and MARÍN-DIAZARAQUE, J. (2002), “A Matrix Variate Generalization of the Power Exponential Family of Distributions”, Communications in Statistics -Theory and Methods, 31(12), 2167–2182.
SERFLING, R. (1980), Approximation Theorems of Mathematical Statistics, Wiley Online Library.
SERFLING, R. (2002), “A Depth Function and a Scale Curve Based on Spatial Quantiles”, in Statistical Data Analysis Based on the L1-Norm and Related Methods, ed. Y. Dodge, pp. 25–38.
SINGH, K. (1991), “A Notion of Majority Depth”, unpublished document.
SMALL, C. (1990), “A Survey of Multidimensional Medians”, International Statistical Review, 58(3), 263–277.
STAHEL, W. (1981), “Robust Estimation: Infinitesimal Optimality and Covariance Matrix Estimators”, unpublished doctoral dissertation, ETH, Zurich, Switzerland.
TUKEY, J. (1975), “Mathematics and the Picturing of Data”, Proceedings of the International Congress of Mathematicians, 2, 523–531.
WHITTLE, P., KAY, J., HAND, D., TARASSENKO, L., BROWN, P., TITTERINGTON, D., TAYLOR, C., GILKS, W., CRITCHLEY, F., MAYNE, A., et al. (1994), “Neural Networks and Related Methods for Classification-Discussion”, Journal of the Royal Statistical Society Series B, 56(3), 437–456.
YEH, I.-C., YANG, K.-J., and TING, T.-M. (2009), “Knowledge Discovery on rfm Model Using Bernoulli Sequence”, Expert Systems with Applications, 36(3), 5866–5871.
ZUO, Y. (2003), “Projection-Based Depth Functions and Associated Medians,” Annals of Statistics, 31(5), 1460–1490.
ZUO, Y. (2006), “Robust Location and Scatter Estimators in Multivariate Analysis”, in Frontiers in Statistics. Dedicated to Peter John Bickel in Honor of his 65th Birthday, Imperial College Press, pp. 467–490.
ZUO, Y., and LAI, S. (2011), “Exact Computation of Bivariate Projection Depth and the Stahel–Donoho Estimator”, Computational Statistics and Data Analysis, 55(3), 1173–1179.
ZUO, Y., and SERFLING, R. (2000a), “General Notions of Statistical Depth Function”, Annals of Statistics, 28(2), 461–482.
ZUO, Y., and SERFLING, R. (2000b), “Structural Properties and Convergence Results for Contours of gns”, Annals of Statistisics, 28(2), 483–499.
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Jiang, Y., Wen, C. & Wang, X. Adaptive Exponential Power Depth with Application to Classification. J Classif 35, 466–480 (2018). https://doi.org/10.1007/s00357-018-9264-z
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DOI: https://doi.org/10.1007/s00357-018-9264-z