Sharp bounds for the maximum of the Goodman-Kruskal \(\tau\) index (Q1979099)

In many situations one variable (say, \(Y\)) is a response variable and the other one (\(Z\)) an explanatory variable, and it is of interest to analyze the dependency of \(Y\) upon \(Z.\) To this aim, \textit{N. Lauro} and \textit{L. D'Ambra} [E. Diday et al. (eds.), Data Analysis and Informatics III, 433-446 (1984)] developed the so called ``non symmetrical correspondence analysis'', a factorial analysis whose aim is to visualize the dependence of \(Y\) on \(Z\) on factorial planes. In this situation, one important index measuring the strength of the dependency of \(Y\) on \(Z,\) namely the Goodman and Kruskal's \(\tau_{Y/Z}\) index, plays a great role. For given marginals, \(\tau\) seldom reaches its theoretical upper bound, \(1.\) It seems therefore reasonable to determine the maximum value the index can assume in a class of contingency tables with given marginals. Three kinds of heuristics developed to this aim are presented in this paper. An upper bound for the maximum is determined so that it is possible to estimate the relative error of the proposed heuristics.