Asymptotic normality

An estimator is asymptotically normal when a rescaled estimation error converges in distribution to a normal law.

Slutsky's theorem combines convergence in distribution with convergence in probability through continuous algebraic operations.

For estimating equations, asymptotic covariance often has the form $A^{-1}B{A}^{-1}$, called a sandwich covariance.

If $\sqrt{n}(T_n-\theta )$ is asymptotically normal, differentiability gives $\sqrt{n}(g(T_n)-g(\theta ))$ the variance multiplied by $g^{\prime }(\theta {)}^2$.

If $r_n(X_n-s)$ converges in distribution to $X$ and $\Phi$ is Hadamard differentiable at $s$ tangentially to a space containing the limit, the displayed transformation rule holds under the usual measurability and tight separable-support conditions. The variables $X_n$ must belong to the domain of $\Phi$.

For a centered bivariate normal sample,This is the multivariate delta method for $g(a,c,b)=c/\sqrt{ab}$.

For an independent sample from $\mathrm{Uniform}[-\theta ,\theta ]$, the maximum likelihood estimator is $M_n={\max }_i|X_i|$ andThe parameter-dependent support makes the convergence rate $n$ rather than $\sqrt{n}$, so the regular maximum-likelihood central limit theorem does not apply.