# Digital Pattern Recognition by K. S. Fu (auth.), Professor King Sun Fu PhD (eds.)

By K. S. Fu (auth.), Professor King Sun Fu PhD (eds.)

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Instit. Statist. Math. 18, 178 (1966) E. A. NADARAYA: Theory Prob. Appl. 10, 186 (1965) J. VAN RYZIN: Ann. Math. Statist 40, 1765 (1969) D. O. LOFTSGAARDEN, C. P. QUESENBERRY: Ann. Math. Statist. 38, 1261 (1965) T. J. WAGNER: IEEE Trans. J. WAGNER: IEEE Trans. STEFANOV: J. Roy. Statist. Soc. 33,1 (1971) G. WAHBA: "Interpolating Spline Methods for Density Estimation II. Variable Knots", Tech. Rep. 337, Dept. of Statistics, University of Wisconsin, Madison, Wisconsin (1973) G. WAHBA: Ann. Statist.

In each of the above procedures symmetry has been sacrificed to obtain a recursive procedure. 1 is that the statistician does not know how to choose the value of h 4. 24] who, for M =2 and known 11: 1 ,11: 2 , examined different ways of choosing h from the data. 4). An argument was given which indicates that this method yields an asymptotically optimal rule. Unfortunately the argument is incomplete since it does not take into consideration the fact that the h chosen is now a random variable. Also discussed was a second method for choosing h, which heuristically has a smaller bias in estimating 11:1 L~ + 1I:2L~.

50]. Density estimation can be viewed as a special case of learning the law of a sequence. 48]. Suppose Xl' X 2, ... is a sequence of independent, identically distributed random vectors with values in IRd and a common probability measure J1. on the Borel u-algebra of~. n of J1. from Xl' ... (B). n is called the empirical probability measure for X b ... (B)I--+O with probability one. 51]. r 32 T. M. COVER and T. J. WAGNER Now, however, assume that we know that Il is absolutely continuous with respect to Lebesgue measure with an almost everywhere continuous probability density f The empirical probability measure seems inappropriate as an approximation to Il since it is atomic with mass lin at X 1, ...