Electronic Journal of Statistics

Monitoring robust regression

Marco Riani, Andrea Cerioli, Anthony C. Atkinson, and Domenico Perrotta

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Robust methods are little applied (although much studied by statisticians). We monitor very robust regression by looking at the behaviour of residuals and test statistics as we smoothly change the robustness of parameter estimation from a breakdown point of 50% to non-robust least squares. The resulting procedure provides insight into the structure of the data including outliers and the presence of more than one population. Monitoring overcomes the hindrances to the routine adoption of robust methods, being informative about the choice between the various robust procedures. Methods tuned to give nominal high efficiency fail with our most complicated example. We find that the most informative analyses come from S estimates combined with Tukey’s biweight or with the optimal $\rho$ functions.

For our major example with 1,949 observations and 13 explanatory variables, we combine robust S estimation with regression using the forward search, so obtaining an understanding of the importance of individual observations, which is missing from standard robust procedures. We discover that the data come from two different populations. They also contain six outliers.

Our analyses are accompanied by numerous graphs. Algebraic results are contained in two appendices, the second of which provides useful new results on the absolute odd moments of elliptically truncated multivariate normal random variables.

Article information

Electron. J. Statist. Volume 8, Number 1 (2014), 646-677.

First available in Project Euclid: 20 May 2014

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62J05: Linear regression 62J20: Diagnostics 62G35: Robustness
Secondary: 62P20: Applications to economics [See also 91Bxx]

Forward search graphical methods least trimmed squares outliers regression diagnostics rho function S estimation truncated normal distribution


Riani, Marco; Cerioli, Andrea; Atkinson, Anthony C.; Perrotta, Domenico. Monitoring robust regression. Electron. J. Statist. 8 (2014), no. 1, 646--677. doi:10.1214/14-EJS897. https://projecteuclid.org/euclid.ejs/1400592267

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