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A dense Gaussian mean lower bound by Assouad’s method

Example

Let $d, n \geq 1$ , $σ > 0$ , and observe $Y \in R^{d}$ with independent coordinates $Y_{i} \sim N (θ_{i}, σ^{2} / n)$ . Estimate $θ$ under squared Euclidean loss over the dense hypercube $Θ = {0, σ / \sqrt{n}}^{d}$ . Then the minimax risk satisfies $\inf_{K} \sup_{θ \in Θ} R_{θ} (K) \geq \frac{σ^{2} d}{16 n} .$

Facts & Assumptions

Given: The finite Gaussian hypercube experiment and an arbitrary randomized estimator $K$ with values in $R^{d}$ .

[F1]

Standard normal and normal laws and Multivariate normal law, including singular covariance: the law $N_{d} (θ, τ^{2} I_{d})$ has independent scalar normal coordinates with means $θ_{i}$ and variance $τ^{2}$ .

[F6]

Statistical experiments and Parameter-measurable statistical models as kernels: the finite parameter cube with its power-set sigma-algebra makes the Gaussian model a measurable probability kernel.

[F2]

KL divergence between equal-variance Gaussian location laws and Additivity of KL divergence for finite product laws: KL divergence between these product laws is $n ∥ θ - θ^{'} ∥_{2}^{2} / (2 σ^{2})$ .

[F3]

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

[F4]

Assouad lower bound for a squared-error hypercube: for coordinate spacing $a$ and largest neighboring-law TV $v$ , squared-error minimax risk is at least $a^{2} d (1 - v) / 8$ .

[F5]

Loss functions and risk, Randomized procedures as probability kernels, Parameter-measurable statistical models as kernels, and Bayes risk and minimax risk: the finite model is a probability-kernel experiment, risk is the kernel integral of squared loss, and minimax risk is the infimum of worst-case risk over procedures.

Proof

1.1F1F2F3F6givenalgebra

Let $a = σ / \sqrt{n}$ , so the parameter set is ${0, a}^{d}$ ; this finite parameter set makes the model a measurable kernel by [F6]. Neighboring vertices differ in one coordinate. By [F2], their KL divergence is $\frac{n a^{2}}{2 σ^{2}} = \frac{1}{2} .$ Pinsker [F3] therefore bounds every neighboring total-variation distance by $\sqrt{(1 / 2) / 2} = 1 / 2$ .

2.1F4F5F6step 1.1algebra∎

Apply [F4] with this spacing and $v \leq 1 / 2$ : $\inf_{K} \sup_{θ \in {0, a}^{d}} R_{θ} (K) \geq \frac{a^{2} d}{8} (1 - \frac{1}{2}) = \frac{σ^{2} d}{16 n} .$ The action and loss are those specified in [F5], which proves the example.

Depends on

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Sources

John Duchi, Lecture Notes on Statistics and Information Theory, Chapter 8, §8.5.3, Example 8.5.3 and §8.8.3 (standard reference, not scraped)