Theorem: Let \(\{ p(x|\theta): \theta \in \Theta \subset \real\}\) be a probability model that is unimodal (with respect to \(\theta\)) and identifiable, and suppose \(X_i \iid p(x|\theta^*)\). Then \(\th \inP \theta^*\).
Proof: Let \(\eps > 0\).
\[\begin{alignat*}{2} & \Pr\{ L(\theta^*) > L(\theta^* - \eps) \} \to 1 &\hspace{4em}& \href{lr-converge.html}{\text{LR convergence}} \,(p \text{ identifiable}) \\ & \Pr\{ L(\theta^*) > L(\theta^* + \eps) \} \to 1 && \href{lr-converge.html}{\text{LR convergence}} \,(p \text{ identifiable}) \\ & \Pr\{ L(\theta^*) > L(\theta^* - \eps) \cap L(\theta^*) > L(\theta^* + \eps)\} \to 1 && \Pr(A \cap B) \ge \Pr(A) + \Pr(B) - 1 \\ & \Pr\{ \th \in (\theta^* - \eps, \theta^* + \eps)\} \to 1 && \text{likelihood not monotone on } N_\eps(\theta^*) \\ & && \implies \th \in N_\eps(\theta^*) \text{ if $p$ } \href{unimodal.html}{\text{unimodal}} \\ \end{alignat*}\]