Gideon Schwarz

On the reciprocity of proximity relations

Abstract The degree of reciprocity of a proximity order is the proportion, P (1), of elements for which the closest neighbor relation is symmetric, and the R value of each element is its rank in the proximity order from its closest neighbor. Assuming a random sampling of points, we show that Euclidean n -spaces produce a very high degree of reciprocity, P(1) ≥ 1 2 , and correspondingly low R values, E ( R ) ≤ 2, for all n . The same bounds also apply to homogeneous graphs, in which the same number of edges meet at every node. Much less reciprocity and higher R values, however, can be attained in finite tree models and in the contrast model in which the “distance” between objects is a linear function of the numbers of their common and distinctive features.

Correlation and the time interval over which the variables are measured

Abstract When two random variables are multiplicative over time, their correlation coefficient is not invariant under changes of the differencing interval even when each of the random variables is a product of i.i.d. variables over time. It is shown that unless Y = kX, k > 0, the coefficient of determination (ϱ2) decreases monotonically as the differencing interval increases, approaching zero in the limit. In sampling for empirical studies, the differencing interval is often selected arbitrarily. Such a choice may dramatically affect the sample correlation coefficient, as well as its statistical significance.

The dark side of the Moebius strip

GIDEON E. SCHWARZ: Born 1933 in Salzburg, Austria. Escaped in 1938, after the Anschluss, to Palestine, today Israel. M.Sc. in Mathematics at the Hebrew University, Jerusalem in 1956. Ph.D. in Mathematical Statistics at Columbia University in 1961. Research fellowships: Miller Institute 1964-66, Institute for Advanced Studies on Mt. Scopus 1975-76. Visiting appointments: Stanford University, Tel Aviv University, University of California in Berkeley. Since 1961, Fellow of the Institute of Mathematical Statistics. Presently, Professor of Statistics at the Hebrew University.

Simple diagnostic tools for inverstigating linear trends in time series

This paper presents a simple diagnostic tool for time series. Based on a coefficient α that veries between 1 and 0, the tool measures the approximation of a time series to an arithmetic progression (i.e., a linear function of time). The proposed α is based on the ratio of the average squared second difference to the average squared first difference of the ginven series. As such, α reduces to the Von Neumann ratio η of the series of first differences, namely, α = 1-η/4. For an arithmetic progression α = 1, and deviations therefrom cause it to decrease. Unlike the correlation coefficient (between the entries and the indics), α is sensitive to local, or piecewise, linearity. Here α is evaluated for an assortment of simple time series models such as random walk, AR(1) and MA(1). Large-sample distribution yields a number of commonly used stochastic models including non-normal process. For most standard deterministic and stochastic models, α stabilizes as n approaches infinity, and provides a statistic that is ...

Game theory and statistics

Publisher Summary Game theory, in particular the theory of two-person zero-sum games, has played a multiple role in statistics. Its principal role has been to provide a unifying framework for the various branches of statistical inference. Statistics, regarded from the game-theoretic point of view, became known as “decision theory.” While unifying statistical inference, decision theory has also proved useful as a tool for weeding out procedures and approaches that have taken hold in statistics without good reason. On a less fundamental level, game theory has contributed to statistical inference the minimax criterion. While the role of this criterion in two-person zero-sum games is central, its application in statistics is problematic. Its justification in game theory is based on the direct opposition of interests between the players, as expressed by the zero-sum assumption. Together with the minimax criterion, randomized, or mixed, strategies also appear in decision theory. The degree of importance of randomization in statistics differs according to which player is randomizing. Mixed strategies for Nature are a priori distributions. In the Bayes approach, these are assumed to represent the Statisticians states of knowledge prior to seeing the data, rather than Natures way of playing the game. Therefore, they are often assumed to be known to the Statistician before he or she makes his or her move, unlike the situation in the typical game-theoretic set-up. Mixed strategies for the Statistician, on the other hand, are, strictly speaking, superfluous from the Bayesian point of view, while according to the minimax criterion, it may be advantageous for the Statistician to randomize, and it is certainly reasonable to grant him or her this option.

The limits in\(\bar d\) of multi-step Markov chainsof multi-step Markov chains

AbstractThe

On attainingđ-đ

\bar d

THE LIMITS IN d OF MULTI-STEP MARKOV CHAINS

of the class of multi-step Markov chains is shown to consist of all direct products of Bernoulli processes with processes of rational pure point spectrum. The class of processes that are approached in