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, 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

Generalizing the OLS and Grid Estimators

The vast majority of market valuations employ either some formal estimator such as ordinary least squares (OLS) or rely upon an informal set of rules defining the grid adjustment estimator. The success of the grid adjustment estimator suggests the data do not obey the ideal assumptions underlying OLS. However, the grid adjustment estimator's lack of a formal statistical foundation makes it difficult to use for inference and other purposes. This article demonstrates how to generalize the grid estimator and OLS to potentially obtain the best features of both. Interestingly, the generalization defines a spatial autoregression. On an empirical example the spatial autoregression outperforms the grid estimator which in turn outperforms OLS. Copyright American Real Estate and Urban Economics Association.