A Revisit to Le Cam’s First Lemma

Le Cam’s first lemma is of fundamental importance to modern theory of statistical inference: it is a key result in the foundation of the Convolution Theorem, which implies a very general form of the optimality of the maximum likelihood estimate and any statistic that is asymptotically equivalent to it. This lemma is also important for developing asymptotically efficient tests. In this note we give a relatively simple but detailed proof of Le Cam’s first lemma. Our proof allows us to grasp the central idea by making analogies between contiguity and absolute continuity, and is particularly attractive when teaching this lemma in a classroom setting.