Location-Scale Depth in Economic Data Stream Analysis

Daniel Kosiorowski

doi:https://doi.org/10.59139/ps.2012.SI1.5

Daniel Kosiorowski Uniwersytet Ekonomiczny w Krakowie Przegląd Statystyczny. Statistical Review, vol. 59, 2012, 1s, pages: 87-108 Published online: 30 June 2012 DOI https://doi.org/10.59139/ps.2012.SI1.5

436 Views 25 Downloads

ARTICLE

(Polish) PDF

ABSTRACT

In this paper we study the properties of the location-scale depth procedures introduced by Mizera & Muller and look into the probabilistic information of the underlying time series model carried by them. We focus our attention on short term multivariate quantile based description of the possible time series model. We study robustness and utility of such the description in a decision making process. In particular we investigate properties of the moving Student median (two dimensional Tukey median in a location–scale problem).

KEYWORDS

Data Stream, Statistical depth function, multivariate median

REFERENCES

[1] Aggerwal Ch.C. (ed.), (2007), Data Streams – Models and Algorithms, Springer, New York.

[2] Ait-Sahalia Y., Jacod J., Li J., (2012), Testing for jumps in noisy high frequency data, Journal of Econometrics, 168, 207-222.

[3] Bocian, Kosiorowski, Węgrzynkiewicz, Zawadzki (2012), pakiet środowiska R {depthproc 1.0} https://r-forge.r-project.org/projects/depthproc/.

[4] Das T., Krishnan S., Venkatasubramanian S., Yi K., (2006), An Information-Theoretic Approach to Detecting Changes in Multi-Dimensional Data Streams. Proceedings of the 38th Symposium on the Interface of Statistics, Computing Science, and Applications (Interface ’06)}, Pasadena, CA.

[5] Fan, J. Yao, Q. (2005), Nonlinear time series: nonparametric and parametric methods, Springer, New York.

[6] Genton M. G., Lucas A. (2003), Comprehensive Definitions of Breakdown Points for Independent and Dependent Observations, Journal of the Royal Statistical Society Series B 65(1), 81-84.

[7] Hall, P., Rodney, C. L. and Yao, Q. (1999). Methods for estimating a conditional distribution function. Journal of the American Statistical Association, 94, (445), 154-163.

[8] Hastie T., Tibshiriani R., Friedman J., (2009), The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Second Edition, Springer.

[9] Hahsler M., Dunhamr H. M., (2010), EMM: Extensible Markov Model for Data Stream Clustering in R, Journal of Statistical Software, 35(5), 2-31.

[10] Huber P., Ronchettii E. M., (2009), Robust Statistics. John Wiley & Sons. New York.

[11] Huber P., (2011) Data Analysis: What Can Be Learned From the Past 50 Years, John Wiley & Sons. New York.

[12] Jacod J., Shiryaev A.N., 2003, Limit Theorems for Stochastic Processes, second ed., Springer-Verlag, New York.

[13] Kong L., Zuo Y., (2010), Smooth Depth Contours Characterize the Underlying Distribution, Journal of Multivariate Analysis 101, 2222-2226.

[14] Kosiorowski D., (2010), Depth Based Procedures for Estimation ARMA and GARCH Models, Y. Lechevallier, G. Saporta (ed.) Proceedings of COMPSTAT’2010 19th International Conference on Computational Statistics, Physica–Verlag, 1207-1214.

[15] Kosiorowski D., (2012), Statystyczne funkcje głębi w odpornej analizie ekonomicznej, Wydawnictwo UEK w Krakowie, Kraków.

[16] Kosiorowski D., (2012), Student depth in robust economic data stream analysis, Colubi A.(Ed.) Proceedings COMPSTAT’2012, The International Statistical Institute/International Association for Statistical Computing.

[17] Kosiorowski D., Snarska M., (2012), Robust monitoring of a multivariate data stream, LINSTAT 2012, artykuł złożony do Communications in Statistics.

[18] Maronna R.A., Martin R.D., Yohai V.J., (2006), Robust Statistics - Theory and Methods. Chichester: John Wiley & Sons Ltd.

[19] Mizera I., (2002), On Depth and Depth Poins: a Calculus. The Annals of Statistics (30), 1681-1736.

[20] Mizera I., C.H. M¨uller (2004), Location-scale Depth (with discussion), Journal of the American Statistical Association 99, 949-966.

[21] Ramsay J.O., Hooker G., Graves S., (2010), Functional Data Analysis with R and Matlab, Springer, New York.

[22] Shalizi C.R., Kontorovich A., (2007), Almost None of the Theory of Stochastic Processes A Course on Random Processes, for Students of Measure-Theoretic Probability, with a View to Applications in Dynamics and Statistics, http://www.stat.cmu.edu/~cshalizi/almost-none/

[23] Serfling R., (2006). Depth Functions in Nonparametric Multivariate Inference, In: Liu R.Y., Serfling R., Souvaine D. L. (Eds.): Series in Discrete Mathematics and Theoretical Computer Science, AMS, vol. 72, 1-15.

[24] Stockis J.-P., Franke J., Kamgaing J.T., (2010). On geometric ergodicity of CHARME models, Journal of the Time Series Analysis 31, 141-152.

[25] Szewczyk W., (2010), Streaming data, Wiley Interdisciplinary Reviews: Computational Statistics, 3(1), (on-line journal).