CoSmo: A Constrained Scatterplot Smoother for Estimating Convex,Monotonic Transformations

In many of the applied sciences, it is common that the forms of empirical relationships are almost completely unknown prior to study. Scatterplot smoothers used in nonparametric regression methods have considerable potential to ease the burden of model specification that a researcher would otherwise face in this situation. Occasionally the researcher will know the sign of the first or second derivatives, or both. This article develops a smoothing method that can incorporate this kind of information. I show that cubic regression splines with bounds on the coefficients offer a simple and effective approximation to monotonic, convex or concave transformations. I also discuss methods for testing whether the constraints should be imposed. Monte Carlo results indicate that this method, dubbed CoSmo, has a lower approximation error than either locally weighted regression or two other constrained smoothing methods. CoSmo has many potential applications and should be especially useful in applied econometrics. As an illustration, I apply CoSmo in a multivariate context to estimate a hedonic price function and to test for concavity in one of the variables.     

Influence analysis based on the case sensitivity function

The case sensitivity function approach to influence analysis is introduced as a natural smooth extension of influence curve methodology in which both the insights of geometry and the power of (convex) analysis are available. In it, perturbation is defined as movement between probability vectors defining weighted empirical distributions. A Euclidean geometry is proposed giving such perturbations both size and direction. The notion of the salience of a perturbation is emphasized. This approach has several benefits. A general probability case weight analysis results. Answers to a number of outstanding questions follow directly. Rescaled versions of the three usual finite sample influence curve measures—seen now to be required for comparability across different-sized subsets of cases—are readily available. These new diagnostics directly measure the salience of the (infinitesimal) perturbations involved. Their essential unity, both within and between subsets, is evident geometrically. Finally it is shown how a relaxation strategy, in which a high dimensional ( O ( ⁿC_m )) discrete problem is replaced by a low dimensional ( O ( n )) continuous problem, can combine with (convex) optimization results to deliver better performance in challenging multiple-case influence problems. Further developments are briefly indicated.