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A Review of Tyler’s Shape Matrix and Its Extensions

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Abstract

In a seminal paper, Tyler (1987a) suggests an M-estimator for shape, which is now known as Tyler’s shape matrix. Tyler’s shape matrix is increasingly popular due to its nice statistical properties. It is distribution free within the class of generalized elliptical distributions. Further, under very mild regularity conditions, it is consistent and asymptotically normally distributed after the usual standardization. Tyler’s shape matrix is still the subject of active research, e.g., in the signal processing literature, which discusses structured and regularized shape matrices. In this article, we review Tyler’s original shape matrix and some recent developments.

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Notes

  1. 1.

    A partial function \(f\!: D\rightarrow C\) is a function from a subset of D to C.

  2. 2.

    See Frahm (2022) for a detailed explanation.

  3. 3.

    Access date: 09.05.2022.

References

  • Abramovich, Y. I., & Spencer, N. K. (2007). Diagonally loaded normalised sample matrix inversion (LNSMI) for outlier-resistant adaptive filtering. In 2007 IEEE International Conference on Acoustics, Speech and Signal Processing – ICASSP ’07 (vol. 3, pp. III–1105–III–1108).

    Google Scholar 

  • Adrover, J. G. (1998). Minimax bias-robust estimation of the dispersion matrix of a multivariate distribution. The Annals of Statistics, 26, 2301–2320.

    Article  MathSciNet  MATH  Google Scholar 

  • Chakraborty, B., Chaudhuri, P., & Oja, H. (1998). Operating transformation retransformation on spatial median and angle test. Statistica Sinica, 8, 767–784.

    MathSciNet  MATH  Google Scholar 

  • Chen, Y., Wiesel, A., & Hero, A. O. (2011). Robust shrinkage estimation of high-dimensional covariance matrices. IEEE Transactions on Signal Processing, 59, 4097–4107.

    Article  MathSciNet  MATH  Google Scholar 

  • Conte, E., De Maio, A., & Ricci, G. (2002). Recursive estimation of the covariance matrix of a compound-gaussian process and its application to adaptive CFAR detection. IEEE Transactions on Signal Processing, 50(8), 1908–1915.

    Article  Google Scholar 

  • Couillet, R., & McKay, M. (2014). Large dimensional analysis and optimization of robust shrinkage covariance matrix estimators. Journal of Multivariate Analysis, 131, 99–120.

    Article  MathSciNet  MATH  Google Scholar 

  • Croux, C., Dehon, C., & Yadine, A. (2010). The k-step spatial sign covariance matrix. Advances in Data Analysis and Classification, 4, 137–150.

    Article  MathSciNet  MATH  Google Scholar 

  • Croux, C., & Haesbroeck, G. (2000). Principal component analysis based on robust estimators of the covariance or correlation matrix: Influence functions and efficiencies. Biometrika, 87, 603–618.

    Article  MathSciNet  MATH  Google Scholar 

  • Dümbgen, L. (1998). On Tyler’s M-functional of scatter in high dimension. Annals of the Institute of Statistical Mathematics, 50(3), 471–491.

    Article  MathSciNet  MATH  Google Scholar 

  • Dümbgen, L., Nordhausen, K., & Schuhmacher, H. (2016). New algorithms for M-estimation of multivariate scatter and location. Journal of Multivariate Analysis, 144, 200–217.

    Article  MathSciNet  MATH  Google Scholar 

  • Dümbgen, L., Pauly, M., & Schweizer, T. (2015). M-functionals of multivariate scatter. Statistics Surveys, 9, 32–105.

    Article  MathSciNet  MATH  Google Scholar 

  • Dümbgen, L., & Tyler, D. E. (2005). On the breakdown properties of some multivariate M-functionals. Scandinavian Journal of Statistics, 32, 247–264.

    Article  MathSciNet  MATH  Google Scholar 

  • Dümbgen, L., & Tyler, D. E. (2016). Geodesic convexity and regularized scatter estimators. arXiv:1607.05455.

    Google Scholar 

  • Fang, K., Kotz, S., & Ng, K. (1990). Symmetric multivariate and related distributions. London: Chapman & Hall.

    Book  MATH  Google Scholar 

  • Frahm, G. (2009). Asymptotic distributions of robust shape matrices and scales. Journal of Multivariate Analysis, 100, 1329–1337.

    Article  MathSciNet  MATH  Google Scholar 

  • Frahm, G. (2022). Power M-estimators for location and scatter. In M. Yi & K. Nordhausen (Eds.), Robust and multivariate statistical methods: Festschrift in Honor of David E. Tyler. Cham: Springer. https://doi.org/10.1007/978-3-031-22687-8_8

  • Frahm, G., & Glombek, K. (2012). Semicircle law of Tyler’s M-estimator for scatter. Statistics & Probability Letters, 82, 959–964.

    Article  MathSciNet  MATH  Google Scholar 

  • Frahm, G., & Jaekel, U. (2010). A generalization of Tyler’s M-estimators to the case of incomplete data. Computational Statistics & Data Analysis, 54, 374–393.

    Article  MathSciNet  MATH  Google Scholar 

  • Frahm, G., & Jaekel, U. (2015). Tyler’s M-estimator in high-dimensional financial-data analysis. In K. Nordhausen, & S. Taskinen (Eds.), Modern nonparametric, robust and multivariate methods: Festschrift in Honour of Hannu Oja (pp. 289–305). Cham: Springer International Publishing.

    Chapter  Google Scholar 

  • Frahm, G., Nordhausen, K., & Oja, H. (2020). M-estimation with incomplete and dependent multivariate data. Journal of Multivariate Analysis, 176, 104569.

    Article  MathSciNet  MATH  Google Scholar 

  • Franks, W. C., & Moitra, A. (2020). Rigorous guarantees for Tyler’s M-estimator via quantum expansion. In J. Abernethy & S. Agarwal (Eds.), Proceedings of Thirty Third Conference on Learning Theory (vol. 125). Proceedings of Machine Learning Research (pp. 1601–1632).

    Google Scholar 

  • Gini, F., & Greco, M. (2002). Covariance matrix estimation for CFAR detection in correlated heavy tailed clutter. Signal Processing, 82, 1847–1859.

    Article  MATH  Google Scholar 

  • Goes, J., Lerman, G., & Nadler, B. (2020). Robust sparse covariance estimation by thresholding Tyler’s M-estimator. The Annals of Statistics, 48, 86–110.

    Article  MathSciNet  MATH  Google Scholar 

  • Hallin, M., Oja, H., & Paindaveine, D. (2006). Semiparametrically efficient rank-based inference for shape. II. Optimal R-estimation of shape. The Annals of Statistics, 34, 2757–2789.

    MathSciNet  MATH  Google Scholar 

  • Hallin, M., & Paindaveine, D. (2006). Semiparametrically efficient rank-based inference for shape. I. optimal rank-based tests for sphericity. The Annals of Statistics, 34, 2707–2756.

    MathSciNet  MATH  Google Scholar 

  • Hettmansperger, T. P., & Randles, R. H. (2002). A practical affine equivariant multivariate median. Biometrika, 89, 851–860.

    Article  MathSciNet  MATH  Google Scholar 

  • Huber, P. J. (1981). Robust statistics. New York: Wiley.

    Book  MATH  Google Scholar 

  • Karoui, N. E. (2009). Concentration of measure and spectra of random matrices: Applications to correlation matrices, elliptical distributions and beyond. The Annals of Applied Probability, 19, 2362–2405.

    Article  MathSciNet  MATH  Google Scholar 

  • Kent, J. T. (1997). Data analysis for shapes and images. Journal of Statistical Planning and Inference, 57, 181–193. Robust Statistics and Data Analysis, Part II.

    Google Scholar 

  • Kent, J. T., & Tyler, D. E. (1988). Maximum likelihood estimation for the wrapped cauchy distribution. Journal of Applied Statistics, 15, 247–254.

    Article  Google Scholar 

  • Maronna, R. A. (1976). Robust M-estimators of multivariate location and scatter. The Annals of Statistics, 4, 51–67.

    Article  MathSciNet  MATH  Google Scholar 

  • Maronna, R. A., Martin, R. D., Yohai, V. J., & Salibian-Barrera, M. (2018). Robust statistics: theory and methods (with R) (2nd ed.). Hoboken: Wiley.

    Book  MATH  Google Scholar 

  • Mériaux, B., Ren, C., Breloy, A., Korso, M. N. E., & Forster, P. (2021). Matched and mismatched estimation of Kronecker product of linearly structured scatter matrices under elliptical distributions. IEEE Transactions on Signal Processing, 69, 603–616.

    Article  MathSciNet  MATH  Google Scholar 

  • Miettinen, J., Nordhausen, K., Taskinen, S., & Tyler, D. E. (2016). On the computation of symmetrized M-estimators of scatter. In C. Agostinelli, A. Basu, P. Filzmoser, & D. Mukherjee (Eds.), Recent advances in robust statistics: Theory and applications (pp. 151–167). New Delhi: Springer India.

    Chapter  Google Scholar 

  • Möttönen, J., Nordhausen, K., & Oja, H. (2010). Asymptotic theory of the spatial median. In J. Antoch, M. Huskova, & P. Sen (Eds.) Nonparametrics and robustness in modern statistical inference and time series analysis: A Festschrift in honor of Professor Jana Jureckova (pp. 182–193). Beachwood, Ohio, USA: Institute of Mathematical Statistics.

    Google Scholar 

  • Möttönen, J., & Oja, H. (1995). Multivariate spatial sign and rank methods. Journal of Nonparametric Statistics, 5, 201–213.

    Article  MathSciNet  MATH  Google Scholar 

  • Nordhausen, K., & Oja, H. (2018a). Independent component analysis: A statistical perspective. WIREs: Computational Statistics, 10, e1440.

    MathSciNet  Google Scholar 

  • Nordhausen, K., & Oja, H. (2018b). Robust nonparametric inference. Annual Review of Statistics and Its Application, 5, 473–500.

    Article  MathSciNet  Google Scholar 

  • Nordhausen, K., & Tyler, D. E. (2015). A cautionary note on robust covariance plug-in methods. Biometrika, 102, 573–588.

    Article  MathSciNet  MATH  Google Scholar 

  • Oja, H. (2010). Multivariate nonparametric methods with R: An approach based on spatial signs and ranks. New York: Springer.

    Book  MATH  Google Scholar 

  • Oja, H., Sirkiä, S., & Eriksson, J. (2006). Scatter matrices and independent component analysis. Australian Journal of Statistics, 35, 175–189.

    Google Scholar 

  • Ollila, E., & Koivunen, V. (2003). Robust antenna array processing using M-estimators of pseudo-covariance. In 14th IEEE Proceedings on Personal, Indoor and Mobile Radio Communications, 2003. PIMRC 2003 (vol. 3, pp. 2659–2663).

    Google Scholar 

  • Ollila, E., Palomar, D. P., & Pascal, F. (2021). Shrinking the eigenvalues of M-estimators of covariance matrix. IEEE Transactions on Signal Processing, 69, 256–269.

    MathSciNet  MATH  Google Scholar 

  • Ollila, E., & Tyler, D. E. (2012). Distribution-free detection under complex elliptically symmetric clutter distribution. In 2012 IEEE 7th sensor array and multichannel signal processing workshop (SAM) (pp. 413–416).

    Google Scholar 

  • Ollila, E., & Tyler, D. E. (2014). Regularized M-estimators of scatter matrix. IEEE Transactions on Signal Processing, 62, 6059–6070.

    Article  MathSciNet  MATH  Google Scholar 

  • Ollila, E., Tyler, D. E., Koivunen, V., & Poor, H. V. (2012). Complex elliptically symmetric distributions: Survey, new results and applications. IEEE Transactions on Signal Processing, 60, 5597–5625.

    Article  MathSciNet  MATH  Google Scholar 

  • Paindaveine, D. (2008). A canonical definition of shape. Statistics & Probability Letters, 78, 2240–2247.

    Article  MathSciNet  MATH  Google Scholar 

  • Paindaveine, D., & Van Bever, G. (2019). Tyler shape depth. Biometrika, 106, 913–927.

    Article  MathSciNet  MATH  Google Scholar 

  • Pascal, F., Chitour, Y., Ovarlez, J.-P., Forster, P., & Larzabal, P. (2008). Covariance structure maximum-likelihood estimates in compound gaussian noise: Existence and algorithm analysis. IEEE Transactions on Signal Processing, 56, 34–48.

    Article  MathSciNet  MATH  Google Scholar 

  • Pascal, F., Chitour, Y., & Quek, Y. (2014). Generalized robust shrinkage estimator and its application to STAP detection problem. IEEE Transactions on Signal Processing, 62, 5640–5651.

    Article  MathSciNet  MATH  Google Scholar 

  • Rublik, F. (2021). On jackknifing the symmetrized Tyler matrix. Statistics, 55, 195–230.

    Article  MathSciNet  MATH  Google Scholar 

  • Salibián-Barrera, M., Aelst, S. V., & Willems, G. (2006). Principal components analysis based on multivariate MM estimators with fast and robust bootstrap. Journal of the American Statistical Association, 101, 1198–1211.

    Article  MathSciNet  MATH  Google Scholar 

  • Schafer, J. (1997). Analysis of Incomplete Multivariate Data. New York: Chapman & Hall.

    Book  MATH  Google Scholar 

  • Sirkiä, S., Taskinen, S., & Oja, H. (2007). Symmetrised M-estimators of multivariate scatter. Journal of Multivariate Analysis, 98, 1611–1629.

    Article  MathSciNet  MATH  Google Scholar 

  • Sirkiä, S., Taskinen, S., Oja, H., & Tyler, D. E. (2009). Tests and estimates of shape based on spatial signs and ranks. Journal of Nonparametric Statistics, 21, 155–176.

    Article  MathSciNet  MATH  Google Scholar 

  • Soloveychik, I., & Trushin, D. (2016). Gaussian and robust Kronecker product covariance estimation: Existence and uniqueness. Journal of Multivariate Analysis, 149, 92–113.

    Article  MathSciNet  MATH  Google Scholar 

  • Soloveychik, I., Trushin, D., & Wiesel, A. (2016). Group symmetric robust covariance estimation. IEEE Transactions on Signal Processing, 64, 244–257.

    Article  MathSciNet  MATH  Google Scholar 

  • Soloveychik, I., & Wiesel, A. (2014). Tyler’s covariance matrix estimator in elliptical models with convex structure. IEEE Transactions on Signal Processing, 62, 5251–5259.

    Article  MathSciNet  MATH  Google Scholar 

  • Soloveychik, I., & Wiesel, A. (2015). Performance analysis of Tyler’s covariance estimator. IEEE Transactions on Signal Processing, 63, 418–426.

    Article  MathSciNet  MATH  Google Scholar 

  • Sun, Y., Babu, P., & Palomar, D. P. (2014). Regularized Tyler’s scatter estimator: Existence, uniqueness, and algorithms. IEEE Transactions on Signal Processing, 62, 5143–5156.

    Article  MathSciNet  MATH  Google Scholar 

  • Sun, Y., Babu, P., & Palomar, D. P. (2015). Robust estimation of structured covariance matrix for heavy-tailed distributions. In 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (pp. 5693–5697).

    Google Scholar 

  • Sun, Y., Babu, P., & Palomar, D. P. (2016). Robust estimation of structured covariance matrix for heavy-tailed elliptical distributions. IEEE Transactions on Signal Processing, 64, 3576–3590.

    Article  MathSciNet  MATH  Google Scholar 

  • Taskinen, S., Croux, C., Kankainen, A., Ollila, E., & Oja, H. (2006). Influence functions and efficiencies of the canonical correlation and vector estimates based on scatter and shape matrices. Journal of Multivariate Analysis, 97, 359–384.

    Article  MathSciNet  MATH  Google Scholar 

  • Taskinen, S., & Oja, H. (2016). Influence functions and efficiencies of k-step Hettmansperger-Randles estimators for multivariate location and regression. In R. Y. Liu & J. W. McKean (Eds.), Robust rank-based and nonparametric methods (pp. 189–207). Cham: Springer International Publishing.

    Chapter  MATH  Google Scholar 

  • Taskinen, S., Sirkiä, S., & Oja, H. (2010). k-Step shape estimators based on spatial signs and ranks. Journal of Statistical Planning and Inference, 140, 3376–3388.

    Article  MathSciNet  MATH  Google Scholar 

  • Tatsuoka, K. S., & Tyler, D. E. (2000). On the uniqueness of S-functionals and M-functionals under nonelliptical distributions. The Annals of Statistics, 28, 1219–1243.

    Article  MathSciNet  MATH  Google Scholar 

  • Tyler, D. E. (1982). Radial estimates and the test for sphericity. Biometrika, 69, 429–436.

    Article  MathSciNet  MATH  Google Scholar 

  • Tyler, D. E. (1983). Robustness and efficiency properties of scatter matrices. Biometrika, 70, 411–420.

    Article  MathSciNet  MATH  Google Scholar 

  • Tyler, D. E. (1987a). A distribution-free M-estimator of multivariate scatter. The Annals of Statistics, 15, 234–251.

    Article  MathSciNet  MATH  Google Scholar 

  • Tyler, D. E. (1987b). Statistical analysis for the angular central Gaussian distribution on the sphere. Biometrika, 74, 579–589.

    Article  MathSciNet  MATH  Google Scholar 

  • Tyler, D. E. (2010). A note on multivariate location and scatter statistics for sparse data sets. Statistics & Probability Letters, 80, 1409–1413.

    Article  MathSciNet  MATH  Google Scholar 

  • Tyler, D. E., & Yi, M. (2020). Breakdown points of penalized and hybrid M-estimators of covariance. arXiv (p. 2003.00078).

    Google Scholar 

  • Wiesel, A. (2012). Geodesic convexity and covariance estimation. IEEE Transactions on Signal Processing, 60, 6182–6189.

    Article  MathSciNet  MATH  Google Scholar 

  • Yang, L., Couillet, R., & McKay, M. (2015). A robust statistics approach to minimum variance portfolio optimization. IEEE Transactions on Signal Processing, 63, 6684–6697.

    Article  MathSciNet  MATH  Google Scholar 

  • Yohai, V. J., & Maronna, R. A. (1990). The maximum bias of robust covariances. Communications in Statistics – Theory and Methods, 19, 3925–3933.

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang, T. (2015). Robust subspace recovery by Tyler’s M-estimator. Information and Inference, 5, 1–21.

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang, T., Cheng, X., & Singer, A. (2016). Marčenko–Pastur law for Tyler’s M-estimator. Journal of Multivariate Analysis, 149, 114–123.

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the editor and referees for their insightful comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland

    Sara Taskinen & Klaus Nordhausen

  2. Department of Mathematics and Statistics, Helmut Schmidt University, Hamburg, Germany

    Gabriel Frahm

  3. Department of Mathematics and Statistics, University of Turku, Turku, Finland

    Hannu Oja

Authors
  1. Sara Taskinen
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  2. Gabriel Frahm
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  3. Klaus Nordhausen
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  4. Hannu Oja
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Correspondence to Sara Taskinen .

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Editors and Affiliations

  1. School of Statistics, Beijing Normal University, Beijing, China

    Mengxi Yi

  2. Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland

    Klaus Nordhausen

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Taskinen, S., Frahm, G., Nordhausen, K., Oja, H. (2023). A Review of Tyler’s Shape Matrix and Its Extensions. In: Yi, M., Nordhausen, K. (eds) Robust and Multivariate Statistical Methods. Springer, Cham. https://doi.org/10.1007/978-3-031-22687-8_2

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