Consistency identifies the correct local experiment. Root-\(n\) estimation then asks whether a procedure tracks every contiguous local alternative with a stable Gaussian error. One-step correction shows that a preliminary estimator can be substantially rougher than root-\(n\) and still be upgraded to efficiency.
Pointwise and uniform consistency
Definition. An estimator \(\tilde\theta_n\) is pointwise consistent if \[ P_\theta\{d(\tilde\theta_n,\theta)>\varepsilon\}\to0 \] for every fixed \(\theta\) and \(\varepsilon>0\). It is uniformly consistent on \(K\subset\Theta\) if \[ \sup_{\theta\in K} P_\theta\{d(\tilde\theta_n,\theta)>\varepsilon\}\to0. \]
Globalization and uniform confidence statements require the second form. Pointwise consistency can coexist with severe nonuniformity near boundaries or weakly identified sequences.
Consistency from uniformly consistent tests
Theorem. Suppose for every \(\varepsilon>0\) there is a finite cover of \(\{\theta:d(\theta,\theta_0)\geq\varepsilon\}\) by sets \(\Theta_{\varepsilon,j}\), and tests \(\varphi_{n,j}\) such that \[ P_{\theta_0}\varphi_{n,j}\to0, \qquad \sup_{\theta\in\Theta_{\varepsilon,j}} P_\theta(1-\varphi_{n,j})\to0. \] Then one can construct an estimator consistent at \(\theta_0\); with uniform versions over \(\theta_0\), a uniformly consistent estimator exists.
This testing construction underlies minimum-distance estimators and posterior consistency. Identifiability alone is qualitative; uniformly powerful tests make it statistical.
Root-\(n\) consistency and regularity
Definition. An estimator is root-\(n\) consistent at \(\theta_0\) if \[ \sqrt n(\hat\theta_n-\theta_0)=O_{P_{\theta_0}}(1). \] It is regular if the centered limit under every \(\theta_0+h/\sqrt n\) is the same law.
Root-\(n\) consistency at one point does not imply regularity. Hodges' estimator is even degenerate at its favored point while failing stability under local alternatives.
One-step correction
Theorem. In an i.i.d. DQM model, let \(\tilde\theta_n\) satisfy \[ \tilde\theta_n-\theta_0=o_p(n^{-1/4}). \] Suppose the score obeys the stochastic expansion \[ \frac1n\sum_{i=1}^n \dot\ell_{\tilde\theta_n}(X_i) = \frac1n\sum_{i=1}^n \dot\ell_{\theta_0}(X_i) -I_{\theta_0}(\tilde\theta_n-\theta_0) +o_p(n^{-1/2}), \] and \(\hat I_n\to_pI_{\theta_0}\). Define \[ \hat\theta_n = \tilde\theta_n+ \hat I_n^{-1} \frac1n\sum_{i=1}^n \dot\ell_{\tilde\theta_n}(X_i). \] Then \[ \sqrt n(\hat\theta_n-\theta_0) = I_{\theta_0}^{-1} \frac1{\sqrt n}\sum_{i=1}^n \dot\ell_{\theta_0}(X_i) +o_p(1). \]
Proof. Substitute the score expansion: \[ \begin{aligned} \hat\theta_n-\theta_0 &= \tilde\theta_n-\theta_0 +\hat I_n^{-1}\left[ \frac1n\sum_i\dot\ell_{\theta_0}(X_i) -I_{\theta_0}(\tilde\theta_n-\theta_0) +o_p(n^{-1/2}) \right]. \end{aligned} \] Consistency of \(\hat I_n\) makes the first and third terms cancel to \(o_p(n^{-1/2})\). Multiply by \(\sqrt n\). \(\square\)
The \(o_p(n^{-1/4})\) preliminary rate is a common sufficient condition because a second-order score remainder is then \[ \|\tilde\theta_n-\theta_0\|^2=o_p(n^{-1/2}). \] With sample splitting and stronger smoothness, related conditions can be weakened or reformulated.
Asymptotic linearity
Definition. An estimator is asymptotically linear if \[ \hat\theta_n-\theta_0 = \frac1n\sum_{i=1}^n\varphi_{\theta_0}(X_i) +o_p(n^{-1/2}), \] where \(P_{\theta_0}\varphi_{\theta_0}=0\) and \(P_{\theta_0}\|\varphi_{\theta_0}\|^2<\infty\).
Proposition. In a regular parametric model, any regular asymptotically linear estimator satisfies \[ P_{\theta_0} \{\varphi_{\theta_0}\dot\ell_{\theta_0}^\top\} =I_p. \] Therefore \[ \operatorname{Var}(\varphi_{\theta_0}) \succeq I_{\theta_0}^{-1}, \] with equality for \(\varphi_{\mathrm{eff}}=I_{\theta_0}^{-1}\dot\ell_{\theta_0}\).
Proof. Le Cam's third lemma says that under local parameter \(h\), the asymptotic mean shift of \(n^{-1/2}\sum\varphi(X_i)\) is \(P(\varphi\dot\ell^\top)h\). Regularity requires this shift to equal \(h\), proving the identity. In the Hilbert space \(L_2(P)\), write \[ \varphi=I^{-1}\dot\ell+r, \qquad P(r\dot\ell^\top)=0. \] Then \[ P(\varphi\varphi^\top)=I^{-1}+P(rr^\top)\succeq I^{-1}. \] \(\square\)
Construction versus lower bounds
LAN and the convolution theorem say what an efficient regular estimator must look like. A one-step estimator constructs it from a preliminary localization and a score. Both halves are needed: a covariance lower bound without a procedure is not attainability, and an asymptotic normal estimator without comparison to the limit experiment is not an efficiency theorem.