Appendix A — Iterative proportional fitting (raking) derivation

This appendix gives the full iterative update rule underlying the raking step summarised in Chapter 3, used to restore the age and tenure marginal targets \mathbf p_A and \mathbf p_T after the associated population’s exponential seeding adjustment (Equation 3.4) disturbs them.

Initialising P^{(0)}_{\text{rake}} = \tilde P, each raking iteration r = 0, 1, 2, \dots performs two rescaling passes,

P^{(r + 1/2)}(a, t) \;=\; P^{(r)}_{\text{rake}}(a, t) \cdot \frac{p_{A_a}}{\sum_{t'} P^{(r)}_{\text{rake}}(a, t')}, \qquad P^{(r+1)}_{\text{rake}}(a, t) \;=\; P^{(r+1/2)}(a, t) \cdot \frac{p_{T_t}}{\sum_{a'} P^{(r+1/2)}(a', t)}, \tag{A.1}

first rescaling every row to sum to p_{A_a}, then every column to sum to p_{T_t}. Repeating this pair of passes for R = 100 iterations (well past convergence for a 5 \times 3 table) yields the raked array P^\star, satisfying

\sum_{t} P^\star(a, t) = p_{A_a} \; \text{ for all } a, \qquad \sum_{a} P^\star(a, t) = p_{T_t} \; \text{ for all } t, \tag{A.2}

to numerical tolerance, while retaining the negative age-renting association introduced by the exponential seeding step – raking restores the specified marginals while preserving the association the adjustment introduced.