A regular statistical model is locally asymptotically normal at when its log-likelihood ratio under local shifts has the expansion
Under local asymptotic normality, a fixed shift has a likelihood ratio limit with . 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.
A Gaussian shift model observes with the shift as parameter. It is the limiting experiment of a locally asymptotically normal model when .
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Local asymptotic normality (LAN) is a concept used in the field of statistics and estimation theory, particularly in the context of statistical inference and asymptotic theory. It provides a framework to analyze the behavior of maximum likelihood estimators (MLEs) and similar statistical procedures in large samples.