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At an interior statistical parameter $\theta$, local asymptotic normality means that there are random vectors ${\Delta }_{n,\theta }$ and a symmetric nonnegative Fisher information matrix $I_{\theta }$ such that, for each fixed $h\in {\mathbb{R}}^p$,A common stronger definition requires this expansion also for $h_n\to h$. The fixed-$h$ version suffices here. A regular identifiable model usually has positive definite Fisher information; nonsingularity is unnecessary for the contiguity conclusion.

Here are sufficient differentiability conditions for a proof by Taylor's theorem. Suppose the local probability density functions have common support, ${\ell }_{\vartheta }(y)=\log f(\vartheta ,y)$ is twice continuously differentiable near $\theta$, differentiation can pass through the normalization integral twice, the score function $s_{\theta }=\nabla {\ell }_{\theta }$ has finite second moment, and the local Hessian matrix is bounded in operator norm by an integrable envelope of a function class. These hypotheses implyThe first identity differentiates $\int f_{\theta }=1$ once; the second uses ${\nabla }^2{f}_{\theta }=f_{\theta }({\nabla }^2{\ell }_{\theta }+s_{\theta }{s}_{\theta }^T)$ and differentiates twice. The central limit theorem gives ${\Delta }_{n,\theta }=n^{-1/2}{\sum }_i{s}_{\theta }(Y_i)\stackrel{d}{\to }{N}_p(0,I_{\theta })$. Taylor's integral remainder givesThe weak law of large numbers and the continuity and integrable-envelope hypotheses make the bracket converge in probability to $-I_{\theta }/2$. Indeed, the expectation of the supremum of the Hessian matrix difference over a shrinking neighborhood tends to zero by dominated convergence, and Markov inequality controls its empirical average. This proves local asymptotic normality and explains the role of both derivatives.

Write $Q_n◃P_n$ for contiguity of probability measures: for every measurable sequence $A_n$, $P_n(A_n)\to 0$ implies $Q_n(A_n)\to 0$. Mutual contiguity requires this in both directions. A useful form of Le Cam's first lemma is the following: if the Radon-Nikodym derivative $L_n$ of the absolutely continuous part of $Q_n$ relative to $P_n$ converges in distribution under $P_n$ to $L$, with $L>0$ almost surely and $\mathbb{E}L=1$, then the two sequences are mutually contiguous. This includes the usual case $L_n=d{Q}_n/d{P}_n$. The mean-one condition ensures uniform integrability of $L_n$ and vanishing mass of any singular part; positivity ensures the reverse implication.

Take $P_n=P_{\theta }^n$ and $Q_n=P_{\theta +h/\sqrt{n}}^n$. Their likelihood ratio on the support of $P_n$ satisfies, by local asymptotic normality and the continuous mapping theorem,This limit is positive and $\mathbb{E}L=e^{-v/2}{e}^{v/2}=1$, including $v=0$. The conditions of Le Cam's first lemma therefore hold, giving mutual contiguity of the product sampling laws. The products, rather than the one-observation laws, are the relevant sequences.

For any $\epsilon >0$, let $A_n=\{\Vert {\hat{\theta }}_n-\theta \Vert >\epsilon \}$. Consistency under $P_{\theta }^n$ makes $P_{\theta }^n(A_n)\to 0$, and contiguity gives $Q_n(A_n)\to 0$. Thus the estimator remains consistent for $\theta$ under each fixed local alternative. Since $\Vert \theta +h/\sqrt{n}-\theta \Vert \to 0$, it is also consistent for the moving parameter in the sense that $\Vert {\hat{\theta }}_n-(\theta +h/\sqrt{n})\Vert \to 0$ in probability under $Q_n$.

For independent identically distributed observations $Y_1,\dots ,Y_n$, the likelihood function and log-likelihood areA maximum-likelihood estimator is a measurable choice ${\hat{\theta }}_n\in {\mathrm{argmax}}_{\theta \in \Theta }{L}_n(\theta )$, when a maximizer exists. It need not be unique. The one-observation score function is $s_{\theta }(y)={\nabla }_{\theta }\log f(y,\theta )$, and the Fisher information matrix isUnder the regularity assumptions, differentiation beneath the integral gives ${\mathbb{E}}_{\theta }{s}_{\theta }={\nabla }_{\theta }\int f=0$. Differentiating again gives the second information identity. Independent observations have total Fisher information $nI(\theta )$.

Assume fixed parameter dimension, an interior true parameter ${\theta }_0$, a positive-definite matrix $I({\theta }_0)$, the standard differentiability and integrability conditions, and statistical consistency of ${\hat{\theta }}_n$. Then the asymptotic normality of a maximum likelihood estimator isHere $I$ is the information per observation. Thus the leading covariance matrix of the estimator itself is $I({\theta }_0{)}^{-1}/n$.

To prove this, statistical consistency places the estimator in an interior ball about ${\theta }_0$ with probability tending to one, where its score function vanishes. Write $d_n={\hat{\theta }}_n-{\theta }_0$ and use the integral first-order Taylor formula for a vector map:This integral matrix is needed in a vector problem; one does not have to assert a common scalar mean-value point for every score component. PutThe regular local uniform law of large numbers, continuity of the expected Hessian matrix, and statistical consistency give $J_n\stackrel{P}{\to }I({\theta }_0)$. The multivariate central limit theorem givesThe limiting information matrix is nonsingular, so $J_n^{-1}\stackrel{P}{\to }I({\theta }_0{)}^{-1}$. Solving the Taylor identity and applying the Slutsky theorem proves the displayed normal limit, since $I^{-1}I{I}^{-1}=I^{-1}$. Boundary parameters or singular information are outside this regular theorem.

Write $\ell (\beta )$ for the log-likelihood and $U=\nabla \ell$ for the score function. Newton method uses the Observed Fisher information $M=-{\nabla }^2\ell$; Fisher scoring uses the Fisher information $M={\mathbb{E}}_{\beta }[-{\nabla }^2\ell ]$.

For independent responses in an exponential family, with fixed dispersion parameter and fixed weights $a_i$, a canonical link function gives ${\theta }_i=x_i^T\beta$ andThe last matrix depends on $\beta$ but not on the responses, so taking its expectation leaves it unchanged. Hence the two updates are identical, wherever this Fisher information matrix is invertible. Equivalently both solve the same iteratively reweighted least squares problem. This statement concerns the regression coefficients at a fixed dispersion parameter; a joint update of an unknown dispersion parameter need not have identical observed and expected full Fisher information matrices.

For one observation with density $f(x,\theta )$, the Fisher information matrix at ${\theta }_0$ isHereSince ${\mathrm{Cov}}_{{\theta }_0}(X)=I_p$, the Fisher information of a multivariate normal location model isThe information in the full independent sample is $n{I}_p$ by Tensorization of Fisher information.

Writing the log-likelihood as $\ell (\beta )=\log L(\beta )$, the Fisher information matrix is the expected negative Hessian matrixUnder the usual regularity conditions it is equivalently the expected outer product of the score function.

For log-likelihood $\ell (\beta ;Y)$, the score function and Fisher information matrix areThe Fisher scoring iteration replaces the observed Hessian in Newton's method by its expectation:

Write $\ell (x,\theta )=\log f(x,\theta )$ and let ${\dot{\ell }}_{{\theta }_0}(x)$ be its score function. A second-order Taylor expansion around ${\theta }_0$ gives, for fixed $h$,where every ${\theta }_{i,n}$ lies between ${\theta }_0$ and ${\theta }_0+h/\sqrt{n}$. The mean-zero score identity, the Fisher information matrix, and the multivariate central limit theorem giveThe assumed regularity and a uniform law of large numbers giveConsequently, by the Slutsky theorem,

For the stated Gaussian shift model, direct expansion of the two normal distribution densities givesSince $I({\theta }_0)X\sim N(0,I({\theta }_0))$, this has exactly the limiting distribution above. Equivalently, both limits are normal with mean $-\frac{1}{2}{h}^{T}I({\theta }_0)h$ and variance $h^{T}I({\theta }_0)h$. This is the local asymptotic normality of the model at ${\theta }_0$.

Let $x_i^T$ be row $i$ of the design matrix, ${\eta }_i=x_i^T\beta =1/{\mu }_i$, and $\gamma =1/\varphi$. The full log-likelihood isDifferentiation gives the score functionandThe Hessian does not depend on $Y$, so the Fisher information matrix is

Letand setThe within-galaxy contrasts $q_k-\bar{q}$ contain no $M_0$, so the parameter-dependent log-likelihood reduces toThe score equations give the maximum-likelihood estimatorsThe calibrator and Hubble-flow samples are independent, so both estimators are unbiased andThe Fisher information matrix for $(M_0,\theta )$ isHence $\mathrm{Var}(\hat{\theta })=({\mathcal{I}}^{-1}{)}_{22}$: $\hat{\theta }$ attains the multiparameter Cramer-Rao bound.

The one-observation score function isIts variance, or equivalently minus the expected second derivative of the log-likelihood, is the Fisher information

Under the usual regularity conditions, consistency and asymptotic normality of a maximum likelihood estimator givewhere $I({\theta }_0)$ is the one-observation Fisher information matrix. In a scalar model this reads $N(0,I({\theta }_0{)}^{-1})$.

Under the standard differentiability, identifiability, moment, and nonsingularity assumptions,Here the one-observation score isand the Fisher information matrix is

Put $K=nY$. Apart from a factor independent of $p$, the likelihood isThus the score function and Fisher information matrix are

The Newton method and Fisher scoring updates are respectivelySince $\hat{p}=Y$, initialization at $p^{(0)}=Y$ requires zero steps for either algorithm. Moreover,so binomial-proportion Fisher scoring reaches the MLE in one step from every interior $p^{(0)}\ne Y$.