Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 37 / 3 / b / Solution

Independence gives the log-likelihoodDifferentiating yields the score equationsWhen a finite maximum-likelihood estimator exists, multiplying these equations by ${\hat{\beta }}^T$ givesThe qualification about a finite estimator matters under complete separation in logistic regression, where coefficient estimates can escape to infinity.