A minimax lower bound is proved by replacing estimation with a testing problem that cannot be solved reliably. Two points give sharp local constants, many-point packings capture metric entropy, and hypercubes exploit coordinatewise structure.
Le Cam's two-point method
Theorem. Let \(d\) be a semimetric and suppose \[ d(\theta_0,\theta_1)\geq2s. \] For every estimator \(\hat\theta\) and \(r>0\), \[ \max_{j=0,1} E_{\theta_j}d(\hat\theta,\theta_j)^r \geq \frac{s^r}{2} \{1-\|P_{\theta_0}-P_{\theta_1}\|_{\mathrm{TV}}\}. \]
Proof. Define a test that chooses the parameter closer to \(\hat\theta\). If it chooses incorrectly under \(\theta_j\), the triangle inequality forces \(d(\hat\theta,\theta_j)\geq s\). Hence \[ E_{\theta_j}d(\hat\theta,\theta_j)^r \geq s^rP_{\theta_j}(\text{testing error}). \] The maximum of two errors is at least half their sum, and the minimum sum of type-I and type-II errors is \(1-\|P_{\theta_0}-P_{\theta_1}\|_{\mathrm{TV}}\). \(\square\)
Hellinger or KL inequalities make the testing affinity calculable: \[ 1-\|P-Q\|_{\mathrm{TV}} \geq1-H(P,Q), \qquad \|P-Q\|_{\mathrm{TV}} \leq\sqrt{\tfrac12D_{\mathrm{KL}}(P\|Q)}. \]
Fano's inequality
Theorem. Let \(J\) be uniform on \(\{1,\ldots,M\}\), and let \(X\mid J=j\sim P_j\). For every classifier \(\hat J(X)\), \[ P(\hat J\ne J) \geq 1-\frac{I(J;X)+\log2}{\log M}. \] Moreover, for any reference law \(Q\), \[ I(J;X) \leq\frac1M\sum_{j=1}^MD_{\mathrm{KL}}(P_j\|Q). \]
Proof. Let \(E=1\{\hat J\ne J\}\). The chain rule for entropy gives \[ H(J\mid X) \leq H(E)+P(E)\log(M-1) \leq\log2+P(E)\log M. \] Since \(I(J;X)=H(J)-H(J\mid X)\) and \(H(J)=\log M\), rearrange. For the second assertion, the mixture law \(\bar P=M^{-1}\sum_jP_j\) minimizes \(M^{-1}\sum_jD_{\mathrm{KL}}(P_j\|Q)\), and \(I(J;X)=M^{-1}\sum_jD_{\mathrm{KL}}(P_j\|\bar P)\). \(\square\)
Corollary (packing lower bound). Suppose \(d(\theta_j,\theta_k)\geq2s\) for \(j\ne k\), and \[ \frac1M\sum_{j=1}^M D_{\mathrm{KL}}(P_{\theta_j}\|P_{\theta_0}) \leq\alpha\log M \] with \(\alpha<1\). Then \[ \inf_{\hat\theta}\sup_{\theta} E_\theta d(\hat\theta,\theta)^r \geq s^r\left( 1-\alpha-\frac{\log2}{\log M} \right). \]
Assouad's hypercube
Theorem (Assouad). Let parameters be indexed by \(\omega\in\{0,1\}^m\). Suppose changing coordinate \(j\) changes the target by at least \(s_j\) in an additive loss, and let \(P_\omega\) be the corresponding laws. Then \[ \inf_{\hat\theta}\sup_\omega E_\omega L(\hat\theta,\theta_\omega) \geq \frac12\sum_{j=1}^m s_j \left[ 1- \left\| \bar P_{j,0}-\bar P_{j,1} \right\|_{\mathrm{TV}} \right], \] where \(\bar P_{j,b}\) averages \(P_\omega\) over vertices with \(\omega_j=b\).
Assouad reduces estimation to \(m\) binary decisions and is especially effective for functions built from many disjoint local perturbations.
Product divergence calculations
Proposition. \[ D_{\mathrm{KL}}\!\left( \bigotimes_iP_i\,\middle\|\,\bigotimes_iQ_i \right) = \sum_iD_{\mathrm{KL}}(P_i\|Q_i), \] while \[ H^2\!\left(\bigotimes_iP_i,\bigotimes_iQ_i\right) \leq\sum_iH^2(P_i,Q_i). \]
These identities make local perturbations tractable. Choose separation large enough to imply meaningful loss, but small enough that total KL or Hellinger information remains bounded.
A nonparametric rate calculation
Example (smooth regression heuristic). Place \(m\asymp h^{-d}\) disjoint bumps of width \(h\) and height \(h^s\) in an \(s\)-smooth function class. Neighboring hypercube vertices differ in squared \(L_2\)-loss by order \(h^{2s+d}\), while \(n\) observations give KL divergence of order \(nh^{2s+d}\) per changed bump. Choose \[ nh^{2s+d}\asymp1, \qquad h\asymp n^{-1/(2s+d)}. \] Summing over \(m\) coordinates yields squared \(L_2\)-risk \[ m h^{2s+d}=h^{2s} \asymp n^{-2s/(2s+d)}. \]
Constrained risk inequalities
A two-point argument can also quantify superefficiency. If an estimator has exceptionally small risk at \(\theta_0\), change-of-measure inequalities force increased risk at a nearby \(\theta_1\). Such constrained risk inequalities make the cost of Hodges-type behavior explicit rather than merely showing failure of regularity.
Limit-experiment lower bounds
Fano and Assouad work directly with finite product laws. Local asymptotic minimax theory instead proves convergence to a limit experiment and imports its Bayes risks. The methods are complementary: direct testing often yields global nonparametric rates, while limit experiments yield sharp local constants and structural conclusions.