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Robust Surveillance of Covariance Matrices Using a Single Observation

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Abstract

In this paper a new technique for monitoring shifts in covariance matrices of Gaussian processes is developed. The processes we monitor are obtained from the covariance matrices estimated using a single observation. These processes follow independent Gaussian distribution in the in-control state, thus allowing for application of standard control charts. Furthermore, in contrary to the existing literature, the suggested procedure is asymptotically robust to the shifts in the mean. The explicit out-of-control distribution for an arbitrary moment of the shift is derived. The performance of numerous multivariate control charts is evaluated in an extensive simulation study and applied to monitoring volatilities on financial markets.

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References

  • Albers, W. and Kallenberg, W. (2004). Are estimated control charts in control? Statistics 38, 67–79.

    Article  MathSciNet  MATH  Google Scholar 

  • Andersson, E., Bock, D. and Frisén, M. (2004). Detection of turning points in business cycles. J. Bus. Cycl. Measur. Anal., 1, 93–108.

    Google Scholar 

  • Azzalini, A. (2005). The skew-normal distribution and related multivariate families. Scand. J. Stat., 32, 159–188.

    Article  MathSciNet  MATH  Google Scholar 

  • Bodnar, O. (2007). Sequential Procedures for Monitoring Covariances of Asset Returns. In Advances in Risk Management (G.N. Gregoriou, ed.). Palgrave, pp. 241–264.

  • Bodnar, O., Bodnar, T. and Okhrin, Y. (2009). Surveillance of the covariance matrix based on the properties of the singular wishart distribution. Comput. Statist. Data Anal., 53, 3372–3385.

    Article  MathSciNet  MATH  Google Scholar 

  • Bodnar, T. and Okhrin, Y. (2008). Properties of the singular, inverse and generalized inverse partitioned Wishart distributions. J. Multivariate Anal., 99, 2389–2405.

    Article  MathSciNet  MATH  Google Scholar 

  • Chan, L.K. and Zhang, J. (2001). Cumulative sum control charts for the covariance matrix. Statist. Sinica, 11, 767–790.

    MathSciNet  MATH  Google Scholar 

  • Conte, S. and De Boor, C. (1981). Elementary numerical analysis. Mc Graw-Hill, London.

    Google Scholar 

  • Crosier, R. (1986). A new two-sided cumulative quality control scheme. Technometrics, 28, 187–194.

    Article  MathSciNet  MATH  Google Scholar 

  • Crosier, R. (1988). Multivariate generalizations of cumulative sum quality-control schemes. Technometrics, 30, 291–303.

    Article  MathSciNet  MATH  Google Scholar 

  • Domínguez-Molina, J., González-Farías, G., Ramos-Quiroga, R. and Gupta, A. (2007). A matrix variate closed skew-normal distribution with applications to stochastic frontier analysis. Comm. Statist. Theory Methods, 36, 1671–1703.

    Article  Google Scholar 

  • Frisén, M. (1992). Evaluations of methods for statistical surveillance. Stat. Med., 11, 1489–1502.

    Article  Google Scholar 

  • Golosnoy, W., Okhrin, I., Ragulin, S. and Schmid, W. (2011). On the application of SPC in finance. Front. Statist. Qual. Contr., 9, 119–132.

    Google Scholar 

  • Gupta, A., Varga, T. and Bodnar, T. (2013). Elliptically contoured models in statistics and portfolio theory. Springer, New-York.

    Book  MATH  Google Scholar 

  • Harville, A. (1997). Matrix algebra from a statistician perspective. Springer, New York.

    Book  MATH  Google Scholar 

  • Hawkins, D. (1981). A CUSUM for a scale parameter. J. Qual. Technol., 13, 228–231.

    Google Scholar 

  • Hawkins, D. (1991). Multivariate quality control based on regressionadjusted variables. Technometrics, 33, 61–75.

    Google Scholar 

  • Hotteling, H. (1947). Multivariate Quality Control – Illustrated by the Air Testing of Sample Bombsights. In Techniques of Statistical Analysis (C. Eisenhart, M.W. Hastay and W. Wallis, eds.). McGraw Hill, pp. 111–184.

  • Huwang, L., Yeh, A. and Wu, C.-W. (2007). Monitoring multivariate process variability for individual observations. J. Qual. Technol., 39, 259–278.

    Google Scholar 

  • Knoth, S. (2003). EWMA schemes with nonhomogenous transition kernel. Sequential Anal., 22, 241–255.

    Article  MathSciNet  MATH  Google Scholar 

  • Kramer, H. and Schmid, W. (2000). The influence of parameter estimation on the ARL of Shewhart-type charts for time series. Statist. Papers, 41, 173–196.

    Article  MathSciNet  MATH  Google Scholar 

  • Lawson, A. and Kleinman, K. (2005). Spatial & syndromic surveillance. Wiley, New-York.

    Book  Google Scholar 

  • Lowry, C., Woodall, W., Champ, C. and Rigdon, S. (1992). A multivariate exponentially weighted moving average control chart. Technometrics, 34, 46–53.

    Article  MATH  Google Scholar 

  • Markowitz, H. (1952). Portfolio selection. J. Financ., 7, 77–91.

    Google Scholar 

  • Messaoud, A., Weihs, C. and Hering, F. (2008). Detection of chatter vibration in a drilling process using multivariate control charts. Comput. Statist. Data Anal., 52, 3208–3219.

    Article  MathSciNet  MATH  Google Scholar 

  • Morais, M. and Pacheco, A. (2001). Some stochastic properties of upper one-sided \(\bar{X}\) and ewma charts for μ in the presence of shifts in σ. Sequential Anal., 20, 1–12.

    Article  MathSciNet  MATH  Google Scholar 

  • Moustakides, G. (1986). Optimal stopping times for detecting changes in distributions. Ann. Statist., 14, 1379–1387.

    Article  MathSciNet  MATH  Google Scholar 

  • Muirhead, R. (1982). Aspects of multivariate statistical theory. Wiley, New York.

    Book  MATH  Google Scholar 

  • Ngai, H. and Zhang, J. (2001). Multivariate cumulative sum control charts based on projection pursuit. Statist. Sinica, 11, 747–766.

    MathSciNet  MATH  Google Scholar 

  • Okhrin, Y. and Schmid, W. (2008). Surveillance of Univariate and Multivariate Linear Time Series. In Financial Surveillance (M. Frisén, ed.). Wiley.

  • Page, P. (1954). Continuous inspection schemes. Biometrika, 41, 100–115.

    Article  MathSciNet  MATH  Google Scholar 

  • Pignatiello, J. and Runger, G. (1990). Comparisons of multivariate CUSUM charts. J. Qual. Technol., 22, 173–186.

    Google Scholar 

  • Pollak, M. and Siegmund, D. (1975). Approximations to the expected sample size of certain sequential tests. Ann. Statist., 3, 1267–1282.

    Article  MathSciNet  MATH  Google Scholar 

  • Reynolds, M.J. and Cho, G.-Y. (2006). Multivariate control charts for monitoring the mean vector and covariance matrix. J. Qual. Technol., 38, 230–253.

    Google Scholar 

  • Roberts, S. (1959). Control chart tests based on geometric moving averages. Technometrics, 1, 239–250.

    Article  Google Scholar 

  • Schipper, S. and Schmid, W. (2001). Sequential methods for detecting changes in the variance of economic time series. Sequential Anal., 20, 235–262.

    Article  MathSciNet  MATH  Google Scholar 

  • Schmid, W. and Tzotchev, D. (2004). Statistical surveillance of the parameters of a one-factor Cox-Ingersoll-Ross model. Sequential Anal., 23, 379–412.

    Article  MathSciNet  MATH  Google Scholar 

  • Shewhart, W. (1931). Economic control of quality of manufactured product. van Nostrand, Toronto.

    Google Scholar 

  • Śliwa, P. and Schmid, W. (2005). Monitoring the cross-covariances of a multivariate time series. Metrika, 61, 89–115.

    Article  MathSciNet  MATH  Google Scholar 

  • Sonesson, C. and Bock, D. (2003). A review and discussion of prospective statistical surveillance in public health. J. Roy. Statist. Soc. A, 166, 5–21.

    Article  MathSciNet  Google Scholar 

  • Srivastava, M. (2003). Singular Wishart and multivariate beta distributions. Ann. Statist. 31, 1537–1560.

    Article  MathSciNet  MATH  Google Scholar 

  • Woodall, W. and Mahmoud, M. (2005). The inertial properties of quality control charts. Technometrics, 47, 425–436.

    Article  MathSciNet  Google Scholar 

  • Woodall, W. and Ncube, M. (1985). Multivariate CUSUM quality control procedures. Technometrics, 27, 285–292.

    Article  MathSciNet  MATH  Google Scholar 

  • Yeh, A., Huwang, L. and Wu, C.-W. (2005). A multivariate EWMA control chart for monitoring process variability with individual observations. IIE Trans., 37, 1023–1035.

    Article  Google Scholar 

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Authors and Affiliations

  1. Physikalisch-Technische Bundesanstalt, Abbestrasse 2-l2, 10587, Berlin, Germany

    Olha Bodnar

  2. Department of Mathematics, Humboldt-University of Berlin, 10099, Berlin, Germany

    Taras Bodnar

  3. Department of Statistics, University of Augsburg, 86159, Augsburg, Germany

    Yarema Okhrin

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  1. Olha Bodnar
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  2. Taras Bodnar
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  3. Yarema Okhrin
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Correspondence to Yarema Okhrin.

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Bodnar, O., Bodnar, T. & Okhrin, Y. Robust Surveillance of Covariance Matrices Using a Single Observation. Sankhya A 76, 219–256 (2014). https://doi.org/10.1007/s13171-013-0044-x

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  • DOI: https://doi.org/10.1007/s13171-013-0044-x

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