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Strong consistency and robustness of the Forward Search estimator of multivariate location and scatter

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  • Cerioli, Andrea
  • Farcomeni, Alessio
  • Riani, Marco

Abstract

The Forward Search is a powerful general method for detecting anomalies in structured data, whose diagnostic power has been shown in many statistical contexts. However, despite the wealth of empirical evidence in favor of the method, only few theoretical properties have been established regarding the resulting estimators. We show that the Forward Search estimators are strongly consistent at the multivariate normal model. We also obtain their finite sample breakdown point. Our results put the Forward Search approach for multivariate data on a solid statistical ground, which formally motivates its use in robust applied statistics. Furthermore, they allow us to compare the Forward Search estimators with other well known multivariate high-breakdown techniques.

Suggested Citation

  • Cerioli, Andrea & Farcomeni, Alessio & Riani, Marco, 2014. "Strong consistency and robustness of the Forward Search estimator of multivariate location and scatter," Journal of Multivariate Analysis, Elsevier, vol. 126(C), pages 167-183.
    • Handle: RePEc:eee:jmvana:v:126:y:2014:i:c:p:167-183
    DOI: 10.1016/j.jmva.2013.12.010
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    3. Torti, Francesca & Corbellini, Aldo & Atkinson, Anthony C., 2021. "fsdaSAS: a package for robust regression for very large datasets including the batch forward search," LSE Research Online Documents on Economics 109895, London School of Economics and Political Science, LSE Library.
    4. Zuppiroli, Marco & Donati, Michele & Riani, Marco & Verga, Giovanni, 2015. "The Impact of Trading Activity in Agricultural Futures Markets," 2015 Fourth Congress, June 11-12, 2015, Ancona, Italy 207848, Italian Association of Agricultural and Applied Economics (AIEAA).
    5. Alessio Farcomeni & Francesco Dotto, 2018. "The power of (extended) monitoring in robust clustering," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 27(4), pages 651-660, December.
    6. Chitradipa Chakraborty & Subhra Sankar Dhar, 2020. "A Test for Multivariate Location Parameter in Elliptical Model Based on Forward Search Method," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 82(1), pages 68-95, February.
    7. Atkinson, Anthony C. & Riani, Marco & Torti, Francesca, 2016. "Robust methods for heteroskedastic regression," Computational Statistics & Data Analysis, Elsevier, vol. 104(C), pages 209-222.
    8. Riani, Marco & Atkinson, Anthony Curtis & Corbellini, Aldo & Farcomeni, Alessio & Laurini, Fabrizio, 2024. "Information Criteria for Outlier Detection Avoiding Arbitrary Significance Levels," Econometrics and Statistics, Elsevier, vol. 29(C), pages 189-205.
    9. Silvia Salini & Andrea Cerioli & Fabrizio Laurini & Marco Riani, 2016. "Reliable Robust Regression Diagnostics," International Statistical Review, International Statistical Institute, vol. 84(1), pages 99-127, April.
    10. Andrea Cerioli & Marco Riani & Anthony C. Atkinson & Aldo Corbellini, 2018. "The power of monitoring: how to make the most of a contaminated multivariate sample," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 27(4), pages 559-587, December.
    11. Arismendi, Juan C. & Broda, Simon, 2017. "Multivariate elliptical truncated moments," Journal of Multivariate Analysis, Elsevier, vol. 157(C), pages 29-44.
    12. Chakraborty, Chitradipa, 2019. "Testing multivariate scatter parameter in elliptical model based on forward search method," Statistics & Probability Letters, Elsevier, vol. 147(C), pages 66-72.
    13. Baishuai Zuo & Chuancun Yin, 2022. "Multivariate doubly truncated moments for generalized skew-elliptical distributions with application to multivariate tail conditional risk measures," Papers 2203.00839, arXiv.org.
    14. Baishuai Zuo & Chuancun Yin & Jing Yao, 2023. "Multivariate range Value-at-Risk and covariance risk measures for elliptical and log-elliptical distributions," Papers 2305.09097, arXiv.org.
    15. Stephen Babos & Andreas Artemiou, 2021. "Cumulative Median Estimation for Sufficient Dimension Reduction," Stats, MDPI, vol. 4(1), pages 1-8, February.
    16. Brenton R. Clarke & Andrew Grose, 2023. "A further study comparing forward search multivariate outlier methods including ATLA with an application to clustering," Statistical Papers, Springer, vol. 64(2), pages 395-420, April.

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