Kolmogorov structure function

Abstract: Kolmogorov's goodness-of-fit measure, Dn , for a sample CDF has consistently been set aside for methods such as the D+n or D-n of Smirnov, primarily, it seems, because of the difficulty of computing the distribution of Dn . As far as we know, no easy way to compute that distribution has ever been provided in the 70+ years since Kolmogorov's fundamental paper. We provide one here, a C procedure that provides Pr(Dn .999 with n's of several thousand, we provide a quick approximation that gives accuracy to the 7th digit for such cases.

Abstract: In 1964 Kolmogorov introduced the concept of the complexity of a finite object (for instance, the words in a certain alphabet). He defined complexity as the minimum number of binary signs containing all the information about a given object that are sufficient for its recovery (decoding). This definition depends essentially on the method of decoding. However, by means of the general theory of algorithms, Kolmogorov was able to give an invariant (universal) definition of complexity. Related concepts were investigated by Solomonoff (U.S.A.) and Markov. Using the concept of complexity, Kolmogorov gave definitions of the quantity of information in finite objects and of the concept of a random sequence (which was then defined more precisely by Martin-Lof). Afterwards, this circle of questions developed rapidly. In particular, an interesting development took place of the ideas of Markov on the application of the concept of complexity to the study of quantitative questions in the theory of algorithms. The present article is a survey of the fundamental results connected with the brief remarks above.

Abstract: Asymptotic theorems on the difference between the (empirical) distribution function calculated from a sample and the true distribution function governing the sampling process are well known. Simple proofs of an elementary nature have been obtained for the basic theorems of Komogorov and Smirnov by Feller, but even these proofs conceal to some extent, in their emphasis on elementary methodology, the naturalness of the results (qualitatively at least), and their mutual relations. Feller suggested that the author publish his own approach (which had also been used by Kac), which does not have these disadvantages, although rather deep analysis would be necessary for its rigorous justification. The approach is therefore presented (at one critical point) as heuristic reasoning which leads to results in investigations of this kind, even though the easiest proofs may use entirely different methods. No calculations are required to obtain the qualitative results, that is the existence of limiting distributions for large samples of various measures of the discrepancy between empirical and true distribution functions. The numerical evaluation of these limiting distributions requires certain results concerning the Brownian movement stochastic process and its relation to other Gaussian processes which will be derived in the Appendix.