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SCOTT: IEEE Trans. Information Theory IT-17, 440 (1971) W. H. ROGERS, T. J. WAGNER: Ann. Statist (1974), submitted J. KIEFER, J. WOLFOWITZ: Trans. Am. Math. Soc. COVER: IEEE Trans. FoLEY: IEEE Trans. J. WAGNER: Ann. Statist 1, 359 (1973) K. FUKUNAGA, D. KESSELL: IEEE Trans. HOSTETLER: IEEE Trans. Information Theory, IT-21, 285 (1975) C. T. WOLVERTON: IEEE Trans. J. WEGMAN: Technometrics 14, 533 (1972) E. 1. WEGMAN: J. Statist. COVER: A Hierarchy of Probability Density Function Estimates, in Frontiers of Pattern Recognition, ed.

Let A be a universal partial recursive function. Definition. KA(fID)= min l(p). 19) A(p,x) = f(x) XED We assume throughout that the inputs (p, n) and (p, x) for A are suitably presented to the computer (partial recursive function) A. Thus the complexity of J given the domain D is the minimum length program p such that a Turing machine A, or equivalently a mechanical algorithm A, can compute J(x) in finite time, for each XED. Example j. Let D = {O, 1}d. Let J(x)= {1, 0, LX;= odd LX;= even. Then K(fID)=c, where c is some small constant independent of d.

KOPLOWlTZ: IEEE Trans. COVER, P. HIRSCHLER: Ann. Statist 3, 939 (1975) C. T. A. ROBERTS: IEEE Trans. Information Theory IT-20, 440 (1974) T. J. WAGNER: IEEE Trans. BooRSTYN: "Detection with Time-Varying Finite-Memory Receivers", Abstracts of papers, 1972 IEEE Intern. Symp. Information Theory, (1972) R. A. ROBERTS, J. R. TOOLEY: IEEE Trans. RoBERTS: IEEE Trans. Information Theory IT-19, 631 (1973) J. R. TOOLEY, R. ROBERTS: IEEE Trans. BAXA, L. W. NOLTE: IEEE Trans. HELLMAN: Information and Control (to be published) H.

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