A General Framework for Constrained Smoothing

There are a wide array of smoothing methods available for finding structure in data. A general framework is developed which shows that many of these can be viewed as a projection of the data, with respect to appropriate norms. The underlying vector space is an unusually large product space, which allows inclusion of a wide range of smoothers in our setup (including many methods not typically considered to be projections). We give several applications of this simple geometric interpretation of smoothing. A major payoff is the natural and computationally frugal incorporation of constraints. Our point of view also motivates new estimates and it helps to understand the finite sample and asymptotic behaviour of these estimates

Bandwidth Selection in Kernel Density Estimation: A Review

Allthough nonparametric kernel density estimation is nowadays a standard technique in explorative data--analysis, there is still a big dispute on how to assess the quality of the estimate and which choice of bandwidth is optimal. The main argument is on whether one should use the Integrated Squared Error or the Mean Integrated Squared Error to define the optimal bandwidth. In the last years a lot of research was done to develop bandwidth selection methods which try to estimate the optimal bandwidth obtained by either of this error criterion. This paper summarizes the most important arguments for each criterion and gives an overview over the existing bandwidth selection methods. We also summarize the small sample behaviour of these methods as assessed in several Monte--Carlo studies. These Monte--Carlo studies are all restricted to very small sample sizes due to the fact that the numerical effort of estimating the optimal bandwidth by any of these bandwidth selection methods is proporti..

Fast Implementation of Density-Weighted Average Derivative Estimation

Given random variables X 2 IR d and Y such that E[Y jX = x] = m(x), the average derivative ffi 0 is defined as ffi 0 = E[rm(X)], i.e., as the expected value of the gradient of the regression function. Average derivative estimation has several applications in econometric theory (Stoker, 1992) and thus it is crucial to have a fast implementation of this estimator for practical purposes. We present such an implementation for a variation known as density-weighted average derivative estimation. This algorithm is based on the ideas of binning or Weighted Averaging of Rounded Points (WARPing). The basic idea of this method is to discretize the original data into a d-variate histogram and to replace in the nonparametric smoothing steps the actual observations by the appropriate bincenters. The non-parametric smoothing steps become thus a (multi-dimensional) convolution between the (discretized) data and the (discretized) smoothing kernel. A Monte-Carlo study demonstrates that with this binne..