The Fiducial Limits of Tetrad-Rank Frequencies.

EINSTEIN’S equations (1936) yield maximum likelihood estimates of the w regional frequency distribution of crossovers in the tetrads from which a given sample of strands was derived, one per tetrad. But clearly, as with all estimates of universe parameters from sample statistics, other values within a certain range cannot be rejected; that range can be ascertained, for each tetrad-rank separately, from the standard error computed by a method also given by WEINSTEIN. The standard errors of the separate frequencies do not, however, define the range of non-rejectable values of the rank frequencies jointly, where there are more than two ranks. It is with the calculation of the joint range that the present note is concerned. Only one special case will be considered here: the frequencies of tetrads of rank 0, 1, 2, . . , k (or, further on, k + 1) with regard not to the regional location of the crossovers but only to the total number per tetrad, estimated from (k + 1) observed strand-rank frequencies, assuming random recurrence and no sister-strand crossing over. All possible sets of (k + 1) tetrad frequencies, expressed in decimal fraction, may be thought of as lying within a k-dimensional, unit-edge hypercube, occupying a “corner” of volume l /k! . Somewhere within the corner is the point representing Weinstein’s maximum likelihood estimate of the universe tetrad-rank frequencies for a given sample. Now consider any other point within the corner, any other set of universe rank frequencies. Around it lie successive hyperellipsoidal shells (with increasingly flattened areas near the surfaces of the corner), each containing a finite set of points representing equally-likely sets of rank frequencies in samples from the given universe. The likelihood decreases outward from shell to shell. The sum of the likelihoods outside a given shell falls from 1 a t the universe point to slightly above 0 at the plane surfaces of the corner (possibly somewhat higher at one of the surfaces). Suppose next that we locate all points within the corner whose equal-likelihood sample shells passing through the maximum likelihood point have some particular outside likelihood sum, P. These points will themselves form a hyperellipsoidal shell, with possibly one or more flattened regions. Our problem is to locate that shell, for P generally somewhere from .01 to .OS. But the complete specifications of such a fiducial limit shell would not be very useful, because incomprehensible in any down-to-earth sense. Obviously then we have to confine our attention to particular points within the shell, to specific sets of