Abstract: In the framework of regression, consider the set of regression submodels (or simply models). By submodel, we mean one or more response variables and a subset of the potential regressors. Imagine the submodels as points insome space. How can we "project" these points onto a (two-dimensional) map so as to visualize and compare them, with intent to isolate a small cluster of submodels having desirable properties? The core of the idea developed here is geometrical. To illustrate this, let us consider the one-response case: each submodel is characterized by two clouds of points: C X C R p - 1 , and C Z ⊂ R p , p - 1 being the number of regressors in the submodel. These clouds will provide the two coordinates of the map: C X . (the regressor cloud) will yield a "complexity" coordinate, and C Z (the all-data cloud) a "lack of fit" coordinate. A submodel is said to be too "complex" if it has too many regressors and/or if its regressors are handicapped by near-linear dependencies (multicollinearity). The measures used for model complexity and lack of fit, have to be validated and mixed in some way. To this aim, we balance complexity and lack of fit by defining a new criterion having the form LCp = log[RSE]+θlog[(p-1)/λ p - 1 ], where θ (0 < θ ≤ 1) is a constant, λ p - 1 is the smallest eigenvalue of the sample correlation matrix of the p - 1 regressors, and RSE is the residual squared error, that is, the sum of the squared residuals for a (p - 1)-regressor submodel.

Abstract: SUMMARY We present asymptotic distributions of the Mallow0s type bounded-influence regression quantile for the linear regression model and also the simultaneous equations model Monte Carlo simulation comparing mean squared errors shows that the bounded-influence one is more ecient than the unbounded-influence one (Koenker and Bassett (1978)) when gross errors occur in the independent-variables-space Analysis of examples involving real data have also been provided