Abstract |
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When stochastic errors are added to data from a distribution with a sharp boundary, such as a changepoint or a frontier, nonparametric estimation of the boundary can be interpreted as a problem of deconvolution. However, that view is not necessarily the most appropriate one, since convergence rates in deconvolution problems are often pathologically slow, particularly when the error distribution is Normal. For this reason we argue that, rather than attempting to estimate the distribution of the uncorrupted data, and thereby approximate the boundary, one might focus more directly on the boundary estimation problem. We suggest estimating the boundary by locating the steepest part of an estimate of the density of the corrupted data. It is shown that the bias intrinsic to this approach is of the same order as the variance, not simply the standard deviation, of the error distribution. This relatively fast rate of convergence in low-noise settings motivates our development of methodology in both univariate and bivariate cases. For example, we show that local linear approximations to the boundary in the univariate case can be used to improve accuracy, by reducing intrinsic bias from the square to the third power of standard deviation. Numerical and theoretical properties of estimators are described. |