Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 4 / 13I / iii / Solution

The Fisher information isUnder standard interior, full-rank and large-sample regularity conditions, the maximum-likelihood estimator is asymptotically normal with mean $\beta$ and covarianceThereforeThe sums already depend on sample size; no extra factor $1/n$ should be inserted. The variance is estimable by substituting fitted means. If all $z_i$ are the same, $ac-b^2=0$ and the two coefficients are not separately identifiable.