 Solution

First distinguish a complete-data calculation, conditional on the missing counts, from optimization of the observed-data likelihood. Introduce death counts $D_j={\sum }_{i=1}^j{n}_{ij}$, 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 areThe survival score is $S_j/{\varphi }_j-D_j/(1-{\varphi }_j)$ 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 thereforeSubstituting the survivor and unobserved-alive totals givesThese 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.