Sire evaluation can logically be formulated as a process of prediction of future progeny of a sire produced by matings with specified females and making their records in some specified environment. Sewall Wright (1931) over 40 years ago suggested three types of prediction that might be of interest: (1) progeny of a particular mating, (2) future daughters in the same herd but out of a new sample of dams, (3) daughters out of a random sample of dams of the breed. Dr. Lush as early as 1931 had elucidated the principles of sire evaluation, Lush (1931 Lush (1933). As was pointed out by Lehman (1961), two types of selection problems have been studied by statisticians. These are Model I and Model II selection, analagous to the corresponding models of analysis of variance. In Model I the candidates for selection are fixed, for example, choices are to be made among treatments, a random sample of observations having been taken on each fixed treatment. No really unified theory has been developed for this type of selection. In contrast, Model II selection involves candidates that are regarded as a random sample from some specified population. Model II represents the classical selection problem in animal breeding and had essentially been solved by Wright and Lush early in the 1930's. Smith's (1936) application to plant breeding and, in particular, Hazel's (1943) application to animal breeding formalized the techniques. A third type of selection that has apparently been overlooked by both statisticians and animal breeders might