A Unifying Analytic Model for Certain Linear Bank Tests. Part I

Summary

A unifying, analytic model is derived for linear rank tests. A somewhat unusual approach is followed in that the derivation proceeds from the grand ensemble of samples whose combined size is N to the particular canonical ensembles of samples of defined size. It is the latter which represent the two‐sample cases of statistical interest, but the concept of a grand ensemble permits to derive a grand partition function which decomposes into the canonical partition functions describing the two‐sample cases of practical interest. More concisely stated, the canonical partition functions represent the degeneracy‐generating functions and hence the probability‐generating functions. As a special case of the grand partition function, a model is obtained for the WILCOXON signed rank test. Of special interest may be the fact that the model is readily adapted to the cases where zero or tied absolute differences obtain. Here too, the entire, exact sampling distributions are easily derived.

The theoretical aspects of the unifying, analytic model are discussed in Part I of this presentation, and the model provides the basis for computational algorithms that are eminently suitable for computer implementations, of the exact tests, in ALGOL, FORTRAN, and APL. Part II of this presentation is devoted to interactive APL implementations of the algorithms, and a number of examples are given. Also, Part II is appended with precision tabulations of the degeneracies, the exact point probabilities, and the exact probabilities of the test statistics.