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Under local asymptotic normality, a fixed $h/\sqrt{n}$ shift has a likelihood ratio limit $\exp (W-v/2)$ with $W\sim N(0,v)$. This is positive with mean one. Le Cam first lemma gives mutual contiguity of the product laws, so consistency at the central parameter transfers to these local alternatives.

If two parameters are separated by at least $2r$ in a metric, any estimator gives a test by deciding whether its distance from the first parameter is below $r$. For $0<\eta <1$, the event $|Z-1|\le \eta$ and Markov inequality give the displayed minimax risk bound, where $Z$ is the product likelihood ratio. The unrestricted claim for every positive $\eta$ is false when both factors on the right are negative.

At level $\alpha$, a most powerful test against a specified simple alternative maximises statistical power among statistical tests whose size of a statistical test is at most $\alpha$. The Neyman-Pearson lemma constructs it by a likelihood ratio threshold. A uniformly most powerful test is most powerful simultaneously against every member of the specified alternative family; maximising power at one alternative alone need not give uniform optimality.

Two sequences of probability laws are mutually contiguous when contiguity of probability measures holds in both directions. A positive mean-one limit of their likelihood ratio is a standard sufficient criterion.