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A relaxed approach to combinatorial problems in robustness and diagnostics

Critchley, Frank; Schyns, Michael; Haesbroeck, Gentiane; Fauconnier, Cecile; Lu, Guobing; Atkinson, Richard A. and Wang, Dong Qian (2010). A relaxed approach to combinatorial problems in robustness and diagnostics. Statistics and Computing, 20(1) pp. 99–115.

DOI (Digital Object Identifier) Link: http://dx.doi.org/10.1007/s11222-009-9119-x
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Abstract

A range of procedures in both robustness and diagnostics require optimisation of a target functional over all subsamples of given size. Whereas such combinatorial problems are extremely difficult to solve exactly, something less than the global optimum can be ‘good enough’ for many practical purposes, as shown by example. Again, a relaxation strategy embeds these discrete, high-dimensional problems in continuous, low-dimensional ones. Overall, nonlinear optimisation methods can be exploited to provide a single, reasonably fast algorithm to handle a wide variety of problems of this kind, thereby providing a certain unity. Four running examples illustrate the approach. On the robustness side, algorithmic approximations to minimum covariance determinant (MCD) and least trimmed squares (LTS) estimation. And, on the diagnostic side, detection of multiple multivariate outliers and global diagnostic use of the likelihood displacement function. This last is developed here as a global complement to Cook’s (in J. R. Stat. Soc. 48:133–169, 1986) local analysis. Appropriate convergence of each branch of the algorithm is guaranteed for any target functional whose relaxed form is—in a natural generalisation of concavity, introduced here—‘gravitational’. Again, its descent strategy can downweight to zero contaminating cases in the starting position. A simulation study shows that, although not optimised for the LTS problem, our general algorithm holds its own with algorithms that are so optimised. An adapted algorithm relaxes the gravitational condition itself.

Item Type: Journal Article
Copyright Holders: 2010 Springer Science+Business Media
ISSN: 0960-3174
Keywords: combinatorial optimisation; concave functions; diagnostics; gravitational functions; nonlinear optimisation; robustness
Academic Unit/Department: Mathematics, Computing and Technology > Mathematics and Statistics
Item ID: 26084
Depositing User: Frank Critchley
Date Deposited: 08 Jan 2011 11:48
Last Modified: 23 Oct 2012 14:30
URI: http://oro.open.ac.uk/id/eprint/26084
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