A Gaussian two-point lower bound

Given: The two-point Gaussian location experiment, with independent observations and arbitrary randomized estimators under squared-error loss.

Standard normal and normal laws and Multivariate normal law, including singular covariance: the observation vector has the product law $N_n(\theta \mathbf{1},{\sigma }^2{I}_n)$, with independent normal coordinates.

Statistical experiments and Parameter-measurable statistical models as kernels: the model assigns one probability law to each parameter; because this parameter set is finite with its power-set sigma-algebra, the assignment is a measurable probability kernel.

KL divergence between equal-variance Gaussian location laws: equal-variance normal location laws satisfy $D(N(a,{\tau }^2)\mathrm{\parallel }N(b,{\tau }^2))=(a-b{)}^2\mathrm{/}(2{\tau }^2)$.

Additivity of KL divergence for finite product laws: for finite product laws, KL divergence is the sum of the factor divergences.

Pinsker’s inequality with event-supremum total variation: under the event-supremum convention, $\mathrm{\parallel }P-Q{\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}}\le \sqrt{D(P\mathrm{\parallel }Q)\mathrm{/}2}$.

Two-point Le Cam lower bound for squared-error risk: two scalar targets separated by $\mathrm{\Delta }$ have minimax squared-error risk at least ${\mathrm{\Delta }}^2(1-\mathrm{\parallel }{P}_0-P_1{\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}})\mathrm{/}8$.

Loss functions and risk, Randomized procedures as probability kernels, and Bayes risk and minimax risk: risk is the kernel expectation of the loss; minimax risk takes the infimum over randomized procedures of their maximum risk.