Two-dimensional projection uniformity for space-filling designs

We investigate a space-filling criterion based on $L_2$-type discrepancies, namely the uniform projection criterion, aiming at improving designs' two-dimensional projection uniformity. Under a general reproducing kernel, we establish a formula for the uniform projection criterion function, which builds a connection between rows and columns of the design. For the commonly used discrepancies, we further use this formula to represent the two-dimensional projection uniformity in terms of the $L_p$-distances of U-type designs. These results generalize existing works and reveal new links between the two seemingly unrelated criteria of projection uniformity and the maximin $L_p$-distance for U-type designs. We also apply the obtained results to study several families of space-filling designs with appealing projection uniformity. Because of good projected space-filling properties, these designs are well adapted for computer experiments, especially for the case where not all the input factors are active.     

The Dynamics of Inequality

The past forty years have seen a rapid rise in top income inequality in the United States. While there is a large number of existing theories of the Pareto tail of the long‐run income distributions, almost none of these address the fast rise in top inequality observed in the data. We show that standard theories, which build on a random growth mechanism, generate transition dynamics that are too slow relative to those observed in the data. We then suggest two parsimonious deviations from the canonical model that can explain such changes: “scale dependence” that may arise from changes in skill prices, and “type dependence,” that is, the presence of some “high‐growth types.” These deviations are consistent with theories in which the increase in top income inequality is driven by the rise of “superstar” entrepreneurs or managers.