Celebratio Mathematica — Rosenblatt

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Works connected to Theodor Gasser

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Smoothing techniques for curve estimation (Heidelberg, 2–4 April 1979). Edited by T. Gasser and M. Rosenblatt. Lecture Notes in Mathematics 757. Springer (Berlin), 1979. MR 564249 Zbl 0407.00010 book

M. Rosenblatt: “Global measures of deviation for kernel and nearest neighbor density estimates,” pp. 181–190 in Smoothing techniques for curve estimation (Heidelberg, 2–4 April 1979). Edited by T. Gasser and M. Rosenblatt. Lecture Notes in Mathematics 757. Springer (Berlin), 1979. MR 564258 Zbl 0417.62030 incollection

A number of estimates of the probability density function (and regression function) have been introduced in the past few decades. The oldest are the kernel estimates and more recently nearest neighbor estimates have attracted attention. Most investigations have dealt with the local behavior of the estimates. There has, however, been some research and some heuristic comment on the utility of global measures of deviation like mean square deviation. Here, it is suggested that in a certain setting such global measures of deviation for kernel estimates may depend far less on tail behavior of the density function than in the case of nearest neighbor estimates. This appears to be due to the unstable behavior of the bias of nearest neighbor density estimates in the tails.