Nonparametric Regression for Threshold Data

Consider a detector which records the times at which the realizations of a nonparametric regression model exceed a certain threshold. If the error distribution is known, the regression function can still be identified from these threshold data. We construct estimators for the regression function. They are transformations of kernel estimators. We determine the bandwidth which minimizes the asymptotic mean average squared error, and construct adaptive estimators for the optimal bandwidth by plug-in methods. Our work is motivated by recent work on stochastic resonance in neuroscience and signal detection theory. In this work it was observed empirically and by simulations that detection of a subthreshold signal is enhanced by the addition of noise, and that there is an optimal noise level. The present work seems to be the first effort to study theoretically how to make best use of this type of threshold data. We compare our model with several models in the literature.