In this article, I breifly explain the idea of Le Cam methods.

Two-Point Argument

The goal is to lower bound the minimax risk

\[R^*(\Theta)=\inf_{\hat\theta}\sup_{\theta\in\Theta}\EE_\theta \ell(\theta,\hat\theta).\]

We first present a version of LeCam’s method as in Lecture notes on: Information-theoretic methods for high-dimensional statistics. Suppose the loss function $\ell:\Theta\times\Theta\to\RR_+$ satisfies the following $\alpha$-triangle inequality

\[\ell(\theta_0,\theta_1)\leq \alpha(\ell(\theta_0,\theta)+\ell(\theta_1,\theta))\]

for some $\alpha>0$, then

\[R^*(\Theta)\geq \sup_{\theta_0,\theta_1\in\Theta} \frac{\ell(\theta_0,\theta_1)}{4\alpha}(1-\text{TV}(\PP_{\theta_0},\PP_{\theta_1})).\]

Interpretation: From Estimation to Testing