Estimating the false negatives due to blocking in record linkage
Section 5. Estimation procedure

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The model parameters may be estimated by maximizing the composite likelihood (Varin, Reid and Firth, 2011) of the sample $n_1,\dots ,n_m.$ For brevity, this composite likelihood is subsequently called likelihood. To develop the EM procedure (Dempster, Laird and Rubin, 1977) it is convenient to first derive the maximum likelihood (ML) equations for the complete data, which are comprised of the latent variables $n_{i|M},$ $n_{i|U}$ and $(c_{i1},\dots ,c_{iG})$ for each $i;$ $c_{ig}$ being the indicator that record $i$ is from class  $g.$

After some algebra, the ML equations for the complete data are as follows.

$$\begin{array}{ccc}{\widehat{p}}_g& =& \frac{{\displaystyle {\sum }_{i=1}^m{c}_{ig}{n}_{i|M}}}{{\displaystyle {\sum }_{i=1}^m{c}_{ig}}}\\ {\widehat{\lambda }}_g& =& \frac{{\displaystyle {\sum }_{i=1}^m{c}_{ig}{n}_{i|U}}}{{\displaystyle {\sum }_{i=1}^m{c}_{ig}}}\\ {\widehat{\alpha }}_g& =& \frac{1}{m}{\displaystyle \sum _{i=1}^m{c}_{ig}}.\end{array}(5.1)$$

Consequently the ML equations for the observed data (the $n_i’\text{s})$ are as follows.

$$\begin{array}{ll}{\widehat{p}}_g\hfill & =\frac{{\displaystyle {\sum }_{i=1}^{m}E[c_{ig}{n}_{i|M}|n_i;\psi ]}}{{\displaystyle {\sum }_{i=1}^{m}E[c_{ig}|n_i;\psi ]}}\hfill \\ {\widehat{\lambda }}_g\hfill & =\frac{{\displaystyle {\sum }_{i=1}^{m}E[c_{ig}{n}_{i|U}|n_i;\psi ]}}{{\displaystyle {\sum }_{i=1}^{m}E[c_{ig}|n_i;\psi ]}}\hfill \\ {\widehat{\alpha }}_g\hfill & =\frac{1}{m}{\displaystyle \sum _{i=1}^{m}E[c_{ig}|n_i;\psi ]}.\hfill \end{array}(5.2)$$

The EM procedure alternates between the M-step given by Equation (5.2) and the E-step equations in Appendix A.

The above procedure may produce consistent point estimators even if it treats the sample $n_1,\dots ,n_m$ as if it were independent and identically distributed. However this is likely to generate some bias when estimating the variance and the critical levels of hypothesis tests.

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