### 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 *i*th postion, and the bullet sign represents element-by-element multiplication.

Expanding $\phi$ in terms of *f* and the basis functions yields

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

## 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