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\Large
\chapter{Some Mathematics}
The calculation of $\hat{p}_{1}(t,g)$ proceeds analogously by calculating
the product of $\hat{p}^{\alpha}_{1}(t,g)$ and $\hat{p}^{\beta}_{1}(t,g)$
using the following formul\ae:

\begin{equation}
   \label{ex-1}
   \hat{p}^{\alpha}_{1} (t,g)    =    1    -
   \frac{ f_{2}    d^{t}_{\left[ 1,2 \right]}    \cdot    g}{
      \beta_{t-1}    -    h    \cdot    f_{1} d^{t-1}_{\left[ 0,1 \right]}    -
         h (1    -    f_{1})    d^{t}_{\left[ 0,1 \right]}}
\end{equation}
and
\begin{equation}
   \label{ex-2}
   \hat{p}^{\beta}_{1} (t,g) = 1 -
   \frac{ (1 - f_{2}) d^{t}_{\left[ 1,2 \right] } \cdot g}{
      \beta_{t-2} - h \cdot f_{1} d^{t-2}_{\left[ 0,1 \right]} -
         (1 - f_{1}) d^{t-1}_{\left[ 0,1 \right] } -
         g \cdot f_{2} d^{t}_{\left[ 1,2 \right]}}
\end{equation}
where $d^{t}_{\left[ 1,2 \right]}$ is the number of registered deaths occurring
in the age interval $\left[ 1,2 \right]$ in the calendar year $t$, and
$f_{2}$ is the fraction of those deaths corresponding to the birth
cohort who reached the first birth in year $t-1$.

As it was the case in earlier equations, the form of equations \ref{ex-1}
and \ref{ex-2} requires the assumption that $h$ and $g$ be constants for
some years before $t$.  In addition, we have assumed that the
completeness factors $h$ and $g$ apply to all deaths occurring within
the age segments $\left[ 0,1 \right]$ and $\left[ 1,2 \right]$
respectively.

If one futher assumes that there is a relation between $h$ and $g$, then
a simple procedure can yield estimates of their actual values.  In
particular, I will assume that

\begin{equation}
   \label{ex-3}
   h = \lambda g
\end{equation}

This assumption suggests that given a value of $x$ one could solve for
$h$ and $g$ through iterations using the relations expressed by
\ref{ex-3} and the required operationalizations expressed by \ref{ex-1}
and \ref{ex-2}.

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