Estimators of a type I geometric distribution from observations on conception times

In order to study distributions of fecundability, Potter and Parker fitted a Pearson Type I geometric distribution (with parametersa andb) to data from the Princeton Fertility Study. They, and subsequently other authors, estimateda andb from the observed moments of the month of first conception. A critical analysis of this method has shown that moment estimators ofa andb are moderately reliable only within a specified range of values ofa. Outside this range, either the estimators are extremely inefficient or their variances are not defined at all. Caution should therefore be taken in adopting this procedure. Furthermore, no moment estimate is defined whena is less than 2. It seems preferable to derive maximum likelihood estimates which have certain optimal properties and are defined for all permissible (i.e. positive) values ofa andb. For large samples, we here present: the covariance matrix (where defined) of the moment estimators, methods of obtaining maximum likelihood estimates and their covariance matrix, and the variances of estimates of specified moments of the fecundability of the sample. Results were obtained for three sets of data; in all cases, the maximum likelihood estimates fit the data better than do the moment estimates. Despite a substantial improvement, however, the fit is still poor for the two sets of data from the Princeton Fertility Study. Possible explanations are: a) that the departures from the assumption of constant fecundability for each couple are sufficient to produce the poor fit, b) that the data are inaccurate, or c) that the method of defining the sample of women from whom the data were obtained resulted in an over-representation of short conception times. The relative importance of these factors is difficult to establish.