Scale space view of curve estimation

Scale space theory from computer vision leads to an interesting and novel a pproach to nonparametric curve estimation. The family of smooth curve estim ates indexed by the smoothing parameter can be represented as a surface cal led the scale space surface. The smoothing parameter here plays the same ro le as that played by the scale of resolution in a visual system. In this pa per, we study in detail various features of that surface from a statistical viewpoint. Pi-Teak convergence of the empirical scale space surface to its theoretical counterpart and some related asymptotic results have been esta blished under appropriate regularity conditions. Our theoretical analysis p rovides new insights into nonparametric smoothing procedures and yields use ful techniques for statistical exploration of features in the data. In part icular, Re have used the scale space approach for the development of an eff ective exploratory data analytic tool called SiZer.