Past exam of the mathematical and Computational Biology course of the University of Cambridge / 2024 / Paper 1 / A1 / d / Solution

Write $s={\sum }_i\log x_i<0$. The score function isSetting it to zero gives the maximum-likelihood estimatorThis lies above $-1$. The observed Fisher information isConsequently the log-likelihood is strictly concave throughout its domain. Its stationary point is the unique global maximum, not merely a candidate local maximum. Indeed the log-likelihood tends to minus infinity as either $\theta \downarrow -1$ or $\theta \to \infty$.