PROBLEM
Given a set of 7 age-group specific fertility rates $_5f_x$ for x=15,20,…45, the objective is to estimate
- average cumulative fertility in the 7 age groups, $_5F_x$
- average years since childbirth in the 7 age groups, $_5Z_x$
Formally, in continuous notation
(1)STRATEGY
The basic idea is to find a fine-grained discrete approximation $\{\phi_i\}$ to the unknown continuous function $f(a)$, and then use matrix operations on the discrete approximation.
Specifically, suppose that B is a 180x7 matrix containing basis functions such that the 180 elements of $\phi = {\bf B} \, f$ correspond to $_{\Delta}f_a$ values for a=10.00, 10.25, …, 54.75. (Call these $a_1 ... a_{180}$ and for this specific case $\Delta = .25$.)
B is derived from some Bayesian posterior ($\phi$ should fit observed data and should have good properties that satisfy priors about a fertility schedule.)
DERIVATION
If we write one of the interior integrals above generically as
(3)or
(4)where $C_{ij}=I(a_i \ge a_j)$, $e_i$ is a column vector of 180 zeroes except for a 1 in the ith postion, and the bullet sign represents element-by-element multiplication.
Expanding $\phi$ in terms of f and the basis functions yields
(6)so that the whole set of $Q_i$ values can be stacked into a vector as
(7)CONVERSION TO AGE-GROUP AVERAGES
We can convert the 180x1 vector back into averages over age groups by approximating the outer integrals in the definitions above.
Define a 7 x 180 grouping matrix
(8)that aggregates fine-grained ages back into averages for age groups 15-19,…45-49.
Then
(9)CUMULATIVE FERTILITY BY AGE GROUP
To integrate fertility up to any specific age, the h function in (3) is always 1, and average cumulative fertility in the 7 standard age groups is
(10)MEAN YEARS SINCE CHILDBEARING BY AGE GROUP
By the same logic, h function in (4) is $h_{ij} = a_i-a_j$, and the numerators for the 7 standard age groups are the elements of
(11)