5fx to 5Fx and 5Zx conversion

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)
\begin{align} _5F_x = \frac{1}{5} \int_{x}^{x+5} \; \int_{0}^{a} \: f(z) dz \; da \end{align}
(2)
\begin{align} _5Z_x = \frac{1}{5} \int_{x}^{x+5} \left( \; \frac{\int_{0}^{a} \: (a-z) \, f(z) dz }{\int_{0}^{a} \: f(z) dz} \right) \; da \end{align}

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)
\begin{align} Q_i \; = \; \int_{0}^{a_i} \: h_i(z) \, f(z) dz \; \approx \; \sum_j I(a_i \ge a_j) \: h_{ij} \, \phi_j \, \Delta \end{align}

or

(4)
\begin{align} Q_i \; \approx \; \sum_j C_{ij} \: h_{ij} \, \phi_j \, \Delta \end{align}
(5)
\begin{align} Q_i \; \approx \; e_i\prime ({\bf C \bullet H}) \; \phi \; \Delta \end{align}

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)
\begin{align} Q_i \; \approx \; e_i\prime ({\bf C \bullet H}) \; {\bf B} \; f \; \Delta \end{align}

so that the whole set of $Q_i$ values can be stacked into a vector as

(7)
\begin{align} Q \; \approx \; ({\bf C \bullet H}) \; {\bf B} \; f \; \Delta \end{align}

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)
\begin{align} G = \frac{1}{20} \; \left( [0 \; I_7 \;0 ] \otimes {1_{20}}\prime \right) \end{align}

that aggregates fine-grained ages back into averages for age groups 15-19,…45-49.

Then

(9)
\begin{align} {\bf G} Q \; \approx \; {\bf G} \: ({\bf C \bullet H}) \; {\bf B} \; f \; \Delta \end{align}

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)
\begin{align} F \approx \; {\bf G} \: {\bf C} \; {\bf B} \; f \; \Delta \end{align}

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)
\begin{align} F \bullet Z \approx \; {\bf G} \: ({\bf C} \bullet {\bf H}) \; {\bf B} \; f \; \Delta \end{align}