Estimating the Upper Support Point in Deconvolution
We consider estimation of the upper boundary point "F"-super- - 1 (1) of a distribution function "F" with finite upper boundary or 'frontier' in deconvolution problems, primarily focusing on deconvolution models where the noise density is decreasing on the positive halfline. Our estimates are based on the (non-parametric) maximum likelihood estimator <formula format="inline"><file name="sjos_545_mu1.gif" type="gif" /></formula> (MLE) of "F". We show that<formula format="inline"><file name="sjos_545_mu2.gif" type="gif" /></formula> (1) is asymptotically never too small. If the convolution kernel has bounded support the estimator <formula format="inline"><file name="sjos_545_mu3.gif" type="gif" /></formula> (1) can generally be expected to be consistent. In this case, we establish a relation between the extreme value index of "F" and the rate of convergence of <formula format="inline"><file name="sjos_545_mu4.gif" type="gif" /></formula> (1) to the upper support point for the 'boxcar' deconvolution model. If the convolution density has unbounded support, <formula format="inline"><file name="sjos_545_mu5.gif" type="gif" /></formula> (1) can be expected to overestimate the upper support point. We define consistent estimators <formula format="inline"><file name="sjos_545_mu6.gif" type="gif" /></formula>, for appropriately chosen vanishing sequences ("&bgr;"<sub>"n"</sub>) and study these in a particular case. Copyright 2007 Board of the Foundation of the Scandinavian Journal of Statistics..
