Maximal structure for the sum of squares of a contingency with fixed margins; a programmed algorithmic solution.

*(French)*Zbl 0639.90101This problem is considered as very difficult - and even quasi-impossible - to resolve. We begin by analyzing the approach that we call “analytic” where a non-exact bound is given by a mathematical formula with respect to the two margins. We show that this approach is necessarily based on an application of the Schwarz inequality, conceived in a logical context. The best bound which can be obtained by this method is in fact less accurate than the one determined from the “best” margin.

The original solution that we propose to construct the optimal configuration and the associated exact bound, is based on the notion of recursive algorithm. The starting point of application of the algorithm is the couple of margins of the contingency table. The most important part of this paper is devoted to the study of this solution. Specific notions are introduced. On the other hand, a mathematical justification of the algorithm is provided as deeply as possible.

Our solution enables the exact and logical normalization of a large family of association coefficients between two nominal qualitative variables. On the other hand, this solution builds an optimal statistical transition between two partitions.

The original solution that we propose to construct the optimal configuration and the associated exact bound, is based on the notion of recursive algorithm. The starting point of application of the algorithm is the couple of margins of the contingency table. The most important part of this paper is devoted to the study of this solution. Specific notions are introduced. On the other hand, a mathematical justification of the algorithm is provided as deeply as possible.

Our solution enables the exact and logical normalization of a large family of association coefficients between two nominal qualitative variables. On the other hand, this solution builds an optimal statistical transition between two partitions.

##### MSC:

90B99 | Operations research and management science |

62H17 | Contingency tables |

90C90 | Applications of mathematical programming |

90C27 | Combinatorial optimization |