The graduation of fertility distributions by polynomial functions

Abstract

There are a number of situations in which it is useful to graduate fertility distributions; for example when the data are obtained from small samples of the population, or when it is desired to translate age-specific fertility rates into ratios of total children ever born per woman in an age group or vice versa. Usually it is convenient to work with rates for five-year age groups of women since the effects of errors due to age mis-statements and chance fluctuations are thus reduced.

This paper presents a method for graduating fertility rates for five-year age groups of women, by means of polynomial functions which are constrained to have the shape required at the limits of reproduction, but otherwise depend on parameters which are estimated from the data. The choice of the number of parameters and the technique for fitting the polynomials will depend on the particular problem. Formule for applying the methods in the most common situations are derived. In all cases the graduated values are obtained as linear combinations of the observed rates with constant multipliers which are determined by the problem. The accuracy of the graduations is indicated and the uses of the formule are illustrated by applications to recorded data.