Local asymptotic normality

Consider an iid family dominated by a common measure, with an interior reference parameter. For simplicity, assume its densities are positive on the same measurable set throughout a parameter neighborhood, ensuring local mutual absolute continuity. Differentiability in quadratic mean means that the square root of the density admits an integrated squared-error linear approximation, with remainder smaller than the squared parameter displacement. Its derivative is one half of the score times the square-root density. Require a mean-zero score with finite second moment and, for the nondegenerate version used here, a positive-definite information matrix. This is a model condition, distinct from smoothness of a numerical objective.

The information matrix is the score covariance. Summing scores and dividing by the square root of sample size gives the central sequence, whose limit is normal by the multivariate central limit theorem. For this iid dominated quadratic-mean differentiable model, the local log likelihood ratio has a linear central-sequence term and a deterministic quadratic correction. The statement holds at each fixed local displacement; stronger uniform assertions over a growing parameter region would require additional work.

The quadratic correction centers the exponential of the Gaussian limit at mean one. It is not an arbitrary penalty inserted into a fitting procedure. Under the local alternative, a suitable change-of-measure argument shifts the central sequence's limiting mean by information times the local displacement, while preserving its covariance. The next lesson proves the relevant transfer principle through contiguity and exponential tilting. Keeping the probability measure explicit avoids confusing the reference distribution with the local-alternative distribution.

A fixed alternative stays a positive distance from the reference parameter and generally lies outside this expansion. Likewise, a normal limit for a maximum-likelihood estimator does not establish quadratic-mean differentiability or LAN for the underlying model. Parameter-dependent support can introduce different scales: estimating a uniform endpoint uses sample extremes and an inverse-sample-size rate. One must examine the model's likelihood geometry rather than assuming that every familiar estimator supplies the same local Gaussian experiment.

An inverse-root-sample-size change is small enough to remain statistically comparable to the reference law, yet large enough to produce nontrivial local power in regular models. Smaller changes are often asymptotically invisible at first order; fixed changes can become perfectly distinguishable. These statements concern regular models at the chosen point, not a universal hierarchy for singular or boundary parameters. Information determines the meaningful local distance, so a displacement should be interpreted together with the model's score variance.

For calculations, first identify the per-observation score and information, then write the central sequence with all variance factors intact. Expand the actual likelihood ratio to verify the sign of the quadratic term. In a normal location model this expansion is exact, which provides a useful calibration for the asymptotic notation. For a Bernoulli family at an interior probability the expansion is approximate, while approaching a degenerate probability may invalidate the fixed-interior argument altogether.