After consistency has localized an \(M\)-estimator, its rate is determined by a competition. The population criterion falls away from its maximum, while the empirical criterion oscillates over a neighborhood. The estimator cannot escape to a radius where deterministic loss uniformly dominates stochastic gain.
Curvature and the local modulus
Definition. For \(M_n(\theta)=P_nm_\theta\), \(M(\theta)=Pm_\theta\), define the local difference class \[ \mathcal M_\delta = \{m_\theta-m_{\theta_0}:d(\theta,\theta_0)\leq\delta\} \] and its stochastic modulus \[ \phi_n(\delta) \geq E^*\|\mathbb G_n\|_{\mathcal M_\delta}. \]
Theorem (rate theorem). Suppose:
- \(\hat\theta_n\to_p\theta_0\) and \[ M_n(\hat\theta_n) \geq M_n(\theta_0)-O_p(\delta_n^\alpha); \]
- for some \(\alpha>0\), \(c>0\), and all nearby \(\theta\), \[ M(\theta)-M(\theta_0) \leq-c\,d(\theta,\theta_0)^\alpha; \]
- \(\phi_n(\delta)/\delta^\beta\) is nonincreasing for some \(0<\beta<\alpha\);
- \(\delta_n\downarrow0\) satisfies \[ \phi_n(\delta_n)\lesssim \sqrt n\,\delta_n^\alpha. \]
Then \[ d(\hat\theta_n,\theta_0)=O_p(\delta_n). \]
Proof. Consider shells \[ S_{j,n} = \{\theta:2^{j-1}\delta_n<d(\theta,\theta_0)\leq2^j\delta_n\}. \] If \(\hat\theta_n\in S_{j,n}\), near-optimality and curvature imply that \[ \sup_{\theta\in S_{j,n}} |\mathbb G_n(m_\theta-m_{\theta_0})| \gtrsim \sqrt n\,2^{j\alpha}\delta_n^\alpha \] up to the smaller optimization error. Markov's inequality and the local maximal bound give probability at most \[ \frac{\phi_n(2^j\delta_n)} {\sqrt n\,2^{j\alpha}\delta_n^\alpha} \leq C2^{-j(\alpha-\beta)} \frac{\phi_n(\delta_n)} {\sqrt n\,\delta_n^\alpha} \lesssim2^{-j(\alpha-\beta)}. \] The tail sum over \(j\geq J\) tends to zero as \(J\to\infty\). Thus \(d(\hat\theta_n,\theta_0)/\delta_n\) is tight. \(\square\)
The balancing equation \[ \phi_n(\delta_n)\asymp\sqrt n\,\delta_n^\alpha \] is the memorable part, but it is only justified after curvature, localization, and the monotonicity needed for peeling have been established.
The regular root-\(n\) regime
Example. Suppose the criterion has quadratic curvature, \[ M(\theta)-M(\theta_0)\leq-c\|\theta-\theta_0\|^2, \] and is \(L_2(P)\)-Lipschitz, \[ \|m_\theta-m_{\theta_0}\|_{P,2} \lesssim\|\theta-\theta_0\|. \] A finite-dimensional entropy bound gives \(\phi_n(\delta)\lesssim\delta\). Balancing \[ \delta_n\asymp\sqrt n\,\delta_n^2 \] yields \(\delta_n=n^{-1/2}\).
Once root-\(n\) localization is known, a second-order Taylor expansion of the population criterion and a linear expansion of the empirical process usually produce asymptotic normality.
Cube-root asymptotics
Example. In many threshold or maximum-score criteria, changing \(\theta\) by distance \(\delta\) changes the indicator only on a set of probability \(O(\delta)\). Hence \[ \|m_\theta-m_{\theta_0}\|_{P,2} \asymp\delta^{1/2} \] and the local empirical modulus is \(\phi_n(\delta)\asymp\delta^{1/2}\). With quadratic population curvature, \[ \delta_n^{1/2} \asymp\sqrt n\,\delta_n^2, \] so \[ \delta_n\asymp n^{-1/3}. \]
The slower rate is not caused by flatter deterministic curvature; it comes from rougher stochastic increments. Pointwise discontinuity makes two nearby parameters differ on a thin region, whose Bernoulli fluctuation has standard deviation proportional to the square root of its probability.
Argmax limits
Theorem (argmax continuous mapping). Let \[ Z_n(h) = a_n\{M_n(\theta_0+r_n^{-1}h)-M_n(\theta_0)\}, \qquad h\in K_n. \] Suppose \(Z_n\rightsquigarrow Z\) locally uniformly on \(\mathbb R^d\), \(Z\) has almost surely a unique maximizer \(\hat h\), and the maximizers \(r_n(\hat\theta_n-\theta_0)\) are tight. Then \[ r_n(\hat\theta_n-\theta_0) \rightsquigarrow\hat h=\arg\max_hZ(h). \]
Tightness comes from the rate theorem. Local process convergence identifies the shape of the limit. Uniqueness prevents the argmax map from being discontinuous at the limiting sample path.
Example (Chernoff-type limit). In one-dimensional cube-root problems the rescaled criterion often converges to \[ Z(h)=\sigma W(h)-c h^2, \] where \(W\) is two-sided Brownian motion. Then \[ n^{1/3}(\hat\theta_n-\theta_0) \rightsquigarrow \arg\max_{h\in\mathbb R}\{\sigma W(h)-ch^2\}. \] Brownian scaling shows this law equals \((\sigma/c)^{2/3}\) times a universal Chernoff random variable, up to the normalization convention.
Entropy-generated nonparametric rates
Proposition. Suppose a local class has entropy \[ \log N(\varepsilon,\mathcal M_\delta,L_2(P)) \lesssim\varepsilon^{-\gamma}, \qquad0<\gamma<2, \] and its \(L_2\)-diameter is order \(\delta\). Chaining gives \[ \phi_n(\delta)\lesssim \int_0^\delta\varepsilon^{-\gamma/2}\,\mathrm{d}\varepsilon \asymp\delta^{1-\gamma/2}. \] Under quadratic curvature, \[ \delta_n\asymp n^{-1/(2+\gamma)}. \]
At the critical exponent \(\gamma=2\), logarithmic factors appear; above it, the global entropy integral diverges and a truncated or model-specific local bound is necessary. Rates should therefore be derived from the actual localized class, not read mechanically from a global smoothness label.