Przemysław Jaśko , Daniel Kosiorowski

(Polish) PDF


In this paper we compare two matrix estimators of multivariate scatter – the minimal covariance determinant estimator MCD with a new proposal an estimator minimizing an incongruence criterion PCS in a context of their applications in economics. We analyze the estimators using simulation studies and using empirical examples related to issues of portfolio building.
In a decision process we often make use of multivariate scatter estimators. Incorrect value of these estimates may result in financial losses. In this paper we compare two robust multivariate scatter estimators – MCD (minimum covariance determinant) and recently proposed PCS (projection congruent subset), which are affine equivariant and have high breakdown points. In the empirical analysis we make use of them in the procedure of weights setting for minimum vari ance and equal risk contribution (ERC) portfolios.


robust estimator of multivariate scatter, MCD, PCS, robust portfolio analysis, realized covariance, minimum risk portfolio, equal risk contribution portfolio


Barndorff-Nielsen O. E., Shephard N., (2004), Econometric Analysis of Realised Covariation: High Frequency Covariance, Regression and Correlation in Financial Economics, Econometrica, 72, 885–925.

Boudt K., Cornelissen J., Payseur S., (2014), Highfrequency: Toolkit for the Analysis of Highfrequency Financial Data in R, URL:

Boudt K., Croux C., Laurent S., (2011), Outlyingness Weighted Covariation, Journal of Financial Econometrics, 9 (4), 657–684.

Butler R. W., Davies P. L., Jhun M., (1993), Asymptotics for the Minimum Covariance Determinant Estimator, The Annals of Statistics, 1385–1400.

Croux C., Haesbroeck G., (1999), Influence Function and Efficiency of the Minimum Covariance Determinant Scatter Matrix Estimator, Journal of Multivariate Analysis, 71 (2), 161–190.

Davies P. L., (1987), Asymptotic Behaviour of S-estimates of Multivariate Location Parameters and Dispersion Matrices, The Annals of Statistics, 1269–1292.

Fiszeder P., (2009), Modele klasy GARCH w empirycznych badaniach finansowych, UMK, Toruń Fleming J., Kirby C., Ostdiek B., (2001), The Economic Value of Volatility Timing, The Journal of Finance, 56 (1), 329–352.

Fleming J., Kirby C., Ostdiek B., (2003), The Economic Value of Volatility Timing Using “Realized” Volatility, Journal of Financial Economics 67, 473–509.

Hautsch N., (2011), Econometrics of Financial High-frequency Data, Springer Science & Business Media.

Hubert M., Rousseeuw P. J., Verdonck T., (2012), A Deterministic Algorithm for Robust Location and Scatter, Journal of Computational and Graphical Statistics, 21 (3), 618–637.

Kosiorowski D., Zawadzki Z., (2014), Selected Issues Related to Online Calculation of Multivariate Robust Measures of Location and Scatter, w: Papież M., Śmiech S., (red.), Proceedings of 8th A. Zeliaś International Conference, Zakopane 2014, 87–96.

Kosiorowski D., (2016), Dilemmas of Robust Analysis of Economic Data Streams, Journal of Mathematical Sciences (Springer), 218 (2), 167–181.

Kosiorowski D., (2015), Two Procedures for Robust Monitoring of Probability Distributions of Economic Data Streams, Operational Research and Decisions, 1, 57–79.

Kosiorowska E., Kosiorowski D. Zawadzki Z., (2015), Evaluation of the Fourth Millenium Development Goal Realisation Using Robust and Nonparametric Tools Offered by a Data Depth Concept, Folia Oeconomica Stietiniensia, 15 (1), 34–52.

Maronna R. A., Martin D., Yohai V., (2006), Robust Statistics, John Wiley & Sons, Chichester.

Pfaff B., (2013), Financial Risk Modelling and Portfolio Optimisation with R, John Wiley & Sons Ltd, London.

Pison G., Van Aelst S., Willems G., (2002), Small Sample Corrections for LTS and MCD, Metrika, 55 (1-2), 111–123.

Pooter M. D., Martens M., Dijk D. V., (2008), Predicting the Daily Covariance Matrix for S&P 100 Stocks Using Intraday Data-but which frequency to use?, Econometric Reviews, 27 (1–3), 199–229.

Schmitt E., Öllerer V., Vakili K., (2014), The Finite Sample Breakdown Point of PCS, Statistics & Probability Letters, 94, 214–220.

Rousseeuw P. J., (1984), Least Median of Squares Regression, Journal of the American Statistical Association, 79, 871–880.

Rousseeuw P. J., Driessen K. V., (1999), A Fast Algorithm for the Minimum Covariance Determinant Estimator, Technometrics, 41 (3), 212–223.

Rousseeuw P., Croux C., Todorov V., Ruckstuhl A., Salibian-Barrera M., Verbeke T., Maechler M., (2015), robustbase: Basic Robust Statistics. R package version 0.92-5. URL

Ryan J. A., Ulrich J. M., (2011), xts: Extensible Time Series. R package version 0.8-2.

Todorov V., Filzmoser P., (2009), An Object-Oriented Framework for Robust Multivariate Analysis, Journal of Statistical Software, 32 (3), 1-47. URL

Vakili K., Schmitt E., (2014), Finding Multivariate Outliers with FastPCS, Computational Statistics & Data Analysis, 69, 54–66.

Wuertz D., Chalabi Y., Chen W., Ellis A., (2009), Portfolio Optimization with R/Rmetrics, Rmetrics eBook, Rmetrics Association and Finance Online, Zurich.

Back to top
© 2019–2022 Copyright by Statistics Poland, some rights reserved. Creative Commons Attribution-ShareAlike 4.0 International Public License (CC BY-SA 4.0) Creative Commons — Attribution-ShareAlike 4.0 International — CC BY-SA 4.0