A relaxed approach to combinatorial problems in robustness and diagnostics

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, lowdimensional 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 (1986) local analysis. Appropriate convergence of each branch of the algorithm is guaranteed for any F. Critchley The Open University, Department of Mathematics and Statistics, United Kingdom E-mail: [email protected] M. Schyns University of Liège, HEC-Management School, QuantOM, Belgium E-mail: [email protected] G. Haesbroeck University of Liège, Department of Mathematics, Belgium E-mail: [email protected] C. Fauconnier, University of Liège, Belgium G. Lu, University of Bristol, United Kingdom R.A. Atkinson, The University of Birmingham, United Kingdom D.Q. Wang, Victoria University of Wellington, United Kingdom 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.