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