= Solution
First distinguish a complete-data calculation, conditional on the missing counts, from optimization of the observed-data likelihood. Introduce death counts $D_j=\sum_{i=1}^jn_{ij}$, survivor counts $S_j=A_{1j}$, observed recapture counts $C_j=\sum_{i=1}^{j-1}m_{ij}$ and unobserved-alive counts $U_j=A_{2j}$. Terms involving these probabilities in the complete-data log likelihood are
$$
\ell=\sum_{j=1}^{J-1}[D_j\log(1-\phi_j)+S_j\log\phi_j]
+\sum_{j=2}^J[C_j\log P_j+U_j\log(1-P_j)]+\text{constant}.
$$
The survival score is $S_j/\phi_j-D_j/(1-\phi_j)$ and the detection score is $C_j/P_j-U_j/(1-P_j)$. Their derivatives are nonpositive, and strictly negative whenever their respective total counts are positive. The interior roots, with boundary interpretations if a success or failure count vanishes, are therefore
$$
\boxed{\widehat\phi_j=\frac{S_j}{S_j+D_j},\qquad
\widehat P_j=\frac{C_j}{C_j+U_j}.}
$$
Substituting the survivor and unobserved-alive totals gives
$$
\widehat\phi_j=
\frac{\sum_{i=1}^j\sum_{s=j+1}^J(m_{is}+n_{is})}
{\sum_{i=1}^j(\sum_{s=j+1}^Jm_{is}+\sum_{s=j}^Jn_{is})},
\quad
\widehat P_j=
\frac{\sum_{i=1}^{j-1}m_{ij}}
{\sum_{i=1}^{j-1}\sum_{s=j}^J(m_{is}+n_{is})}.
$$
These are the requested complete-data maximizing ratios. Here $n_{iJ}$ denotes the terminal missing count discussed in part (b); pre-release cells and nonexistent release cohorts contribute zero. If a denominator is zero, the likelihood is constant in that parameter rather than defining a $0/0$ estimate. Also $P_1$ does not occur in this likelihood and has no identified maximum. With unobserved $n$, these ratios alone do not yet supply an observed-data estimate; the <EM algorithm> below supplies the missing-count expectations.
Back to article page
