Aspects of parameter estimation in ascertainment sampling schemes.

It has recently been suggested that ascertainment sampling estimation procedures commonly used are not fully efficient in that the number of unobserved families is an unknown parameter that should be estimated (contrary to common practice) along with the genetic parameters for fully efficient estimation. It has also been suggested that the frequency distribution of family size contains unknown parameters that should similarly be estimated with the genetic parameters. These two suggestions are considered in this paper. It is shown by means of an equivalence theorem that in both cases the estimates and their variances obtained by adopting the suggested procedure are identical with those found by ignoring the unobserved families and by ignoring the family-size distribution. This demonstration leads to a formal justification of further procedures, in particular: (1) use of "method-of-moments" estimators, (2) ignoring the ascertainment scheme in some cases when estimating parameters, and (3) forming estimates of parameters when various parts of the data are obtained by different ascertainment schemes.