8 Appendix A: A deep dive into the convergence behaviour of MLE estimates

In the course of our work we thought it was particularly interesting to further focus on the actual convergence behavior of the MLE-based estimators. By this, we want to see how the estimators evolve after each iteration.

We will visualize the convergence behavior/the evolution of the estimates up to 1000 iterations. This is executed based on a stratified sample (strata: three level pop.area.kind variable), n = 150). We measure this convergence behavior through the use of MAD and AAD which was explained in chapter 5. We calculate the AAD and MAD for each iteration and see how estimates evolve, as the iterations increase.

• Average Absolute Discrepancy (AAD): \(D_{avg}(u,v) = \frac{1}{U}\sum_{j = 1}^{J} |u_{j} - v_{j}|\)

• Maximum Absolute Discrepancy (MAD): \(D_{max}(\textbf{u,v}) \overset{\underset{\mathrm{def}}{}}{=} \sum_{\mathit{j = 1}}^{\mathit{J}}\left | u_{j} - v_{j} \right |\)