I.3.7.6: Rasch, Once More

The Rasch example has a rather simple structure for the second derivatives of the negative log-likelihood $f$ defined in section rasch.

The elements of $\mathcal{D}_{12}f(x)$ are equal to $\pi_{ij}(x)(1-\pi_{ij}(x))$, while $\mathcal{D}_{11}f(x)$ is a diagonal matrix with the row sums of $\mathcal{D}_{12}f(x),$ and $\mathcal{D}_{22}f(x)$ is a diagonal matrix with the column sums.

This means that computing the eigenvalues of $$[\mathcal{D}_{11}f(x)]^{-1}\mathcal{D}_{12}f(x)[\mathcal{D}_{22}f(x)]^{-1}\mathcal{D}_{21}f(x)$$ amounts, in this case, to a correspondence analysis of the matrix with elements $\pi_{ij}(x)(1-\pi_{ij}(x))$.The speed of convergence will depend on the maximum correlation, i.e. on the degree in which the off-diagonal matrix $\mathcal{D}_{12}f(x)$ deviates from independence.