5.4 Joint maximum likelihood estimation

When assuming all students are independent, the joint likelihood $$\begin{equation} L( {\mathbf{Y}}|{\alpha},\mathbf{g},\mathbf{s})=\prod_{i = 1}^N\prod_{j = 1}^J P ({Y_{ij}}=1|\mathbf{\alpha}_i)^{Y_{ij}}[1 - P(Y_{ij}=1|\mathbf{\alpha}_i)]^{1 - {Y_{ij}}} \end{equation}$$ The joint maximum likelihood estimation (JMLE) finds ${\alpha},\mathbf{g},\mathbf{s}$ that can maximize $L( {\mathbf{Y}}|{\alpha},\mathbf{g},\mathbf{s})$ or its logarithm.

Neyman & Scott (1948) has showed that JMLE produced problematic estimates of parameters related with individuals.

References

Neyman, J., & Scott, E. L. (1948). Consistent estimates based on partially consistent observations. Econometrica, 16, 1–32.