Retrospective Adjustment
One can adjust the initial population numbers at age vector \(\underline{N}(1)\) to reflect a retrospective pattern in calculating these estimates. In this case, the user must determine an appropriate vector of retrospective bias-correction coefficients, denoted by \(\underline{C}\), to apply to the vector \(\underline{N}(1)\). These multiplicative bias-correction coefficients may be age-specific or constant across age classes. The bias-corrected initial population vector \(\underline{N}^*(1)\) is calculated from the element-wise product of \(\underline{N}(1)\) and \(\underline{C}\) as
\[ \underline{N}^*(1) = {\bigl( C_1 \cdot N_1(1), \ \dots,\ C_a \cdot N_a(1),\ \dots,\ C_A \cdot N_A(1) \bigr)^T} \]
Note that the bias-correction coefficients are applied to all initial population vectors. If the bias-correction coefficients are determined to be constant across age classes then \(\underline{C} = (C, C, ..., C)^T\) and the bias-corrected initial population vector is
\[ \underline{N}^*(1) = {\bigl( C \cdot N_1(1), \ \dots,\ C \cdot N_a(1),\ \dots,\ C \cdot N_A(1) \bigr)^T}\ =\ C \cdot\underline{N}(1) \]
The bias-correction coefficients are only applied in the first time period of the projection time horizon to reflect uncertainty in the estimated population size at age. Mohn (1999) provides an informative presentation of the retrospective problem in sequential population analysis.