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Exact worst-case risk of the truncated estimator

Statement

Assume Countable Choice and let $ϵ \geq 0$ . In the Gaussian sequence inverse model with positive singular numbers $(s_{j})_{j \in N_{> 0}}$ , let $Q \geq 0$ and let $(a_{j})_{j \in N_{> 0}}$ be positive and nondecreasing. For each $θ \in Θ_{a} (Q)$ , define the squared sequence risk of the cutoff estimator at $D \in N_{0}$ by $R_{θ} (D) : = E_{θ} ⁣ [∥ {\hat{θ}}_{D} (Z) - θ ∥_{ℓ^{2}}^{2}] .$ Then its exact worst-case squared sequence risk is $\sup_{θ \in Θ_{a} (Q)} R_{θ} (D) = \frac{Q^{2}}{a_{D + 1}^{2}} + ϵ^{2} \sum_{j = 1}^{D} s_{j}^{- 2},$ where the sum is $0$ for $D = 0$ . The supremum is attained by the one-coordinate truth $θ^{⋆} = (Q / a_{D + 1}) e_{D + 1}$ , with $θ^{⋆} = 0$ when $Q = 0$ .

Statement references

Litao Ding and Peter Mathé, Minimax rates for statistical inverse problems under general source conditions, Lemma 1 and its complete proof, PDF pp. 2–3

Iain M. Johnstone, Gaussian estimation: Sequence and wavelet models, §3.2, equations (3.11)–(3.12), printed pp. 60–61

Facts & Assumptions

[F1]

Gaussian sequence inverse model: For $ϵ > 0$ , $Z_{j} = θ_{j} + ϵ s_{j}^{- 1} ξ_{j}$ with $E ξ_{j} = 0$ and $E ξ_{j}^{2} = 1$ ; at $ϵ = 0$ , $Z_{j} = θ_{j}$ deterministically.

[F2]

Weighted ellipsoid class for sequence inversion: $θ \in Θ_{a} (Q)$ means $θ \in ℓ^{2}$ and $\sum_{j \geq 1} a_{j}^{2} θ_{j}^{2} \leq Q^{2}$ , with $0 < a_{1} \leq a_{2} \leq \dots$ .

[F3]

Singular-value cutoff estimator: ${\hat{θ}}_{D, j} (Z) = Z_{j}$ for $1 \leq j \leq D$ and $0$ for $j > D$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is assumed only for the upstream compact SVD coordinate identification; it is not used in the finite-coordinate risk calculation.

Given: Countable Choice, $ϵ \geq 0$ , positive singular numbers $(s_{j})$ , a cutoff $D \in N_{0}$ , and the ellipsoid $Θ_{a} (Q)$ with positive nondecreasing weights.

Proof

1.1givenA1F1F2F3algebra

Under [A1], the model coordinates in [F1] are fixed by the countable-choice SVD branch. If $ϵ > 0$ , [F1] and [F3] give ${\hat{θ}}_{D, j} (Z) - θ_{j} = ϵ s_{j}^{- 1} ξ_{j}$ for $j \leq D$ and $- θ_{j}$ for $j > D$ , hence $∥ {\hat{θ}}_{D} (Z) - θ ∥_{ℓ^{2}}^{2} = ϵ^{2} \sum_{j = 1}^{D} s_{j}^{- 2} ξ_{j}^{2} + \sum_{j > D} θ_{j}^{2}$ . If $ϵ = 0$ , the retained error is zero and the omitted error is $- θ_{j}$ , so the squared error is $\sum_{j > D} θ_{j}^{2}$ . In both cases the error is in $ℓ^{2}$ because the estimator has finite support and $θ \in ℓ^{2}$ .

2.1step 1.1F1algebra

For $ϵ > 0$ , taking expectation in the finite random sum of [step 1.1] and using $E ξ_{j}^{2} = 1$ from [F1] gives $R_{θ} (D) = ϵ^{2} \sum_{j = 1}^{D} s_{j}^{- 2} + \sum_{j > D} θ_{j}^{2}$ . For $ϵ = 0$ , the deterministic error in [step 1.1] gives the same formula with the first term zero. The deterministic tail is finite because $θ \in ℓ^{2}$ .

3.1givenF2step 2.1algebra

For every $θ \in Θ_{a} (Q)$ , monotonicity gives $a_{j} \geq a_{D + 1}$ for $j > D$ , so $\sum_{j > D} θ_{j}^{2} = \sum_{j > D} a_{j}^{2} θ_{j}^{2} a_{j}^{- 2} \leq a_{D + 1}^{- 2} \sum_{j > D} a_{j}^{2} θ_{j}^{2} \leq Q^{2} / a_{D + 1}^{2}$ . Thus [step 2.1] is bounded above by the asserted right-hand side.

4.1F2step 2.1step 3.1constructalgebra

The sequence with $θ_{D + 1}^{⋆} = Q / a_{D + 1}$ and all other coordinates zero belongs to $ℓ^{2}$ and satisfies $\sum_{j \geq 1} a_{j}^{2} (θ_{j}^{⋆})^{2} = Q^{2}$ , including $Q = 0$ . Its tail squared norm is exactly $Q^{2} / a_{D + 1}^{2}$ , so it attains the upper bound from [step 3.1]. The finite-coordinate variance term in [step 2.1] is independent of $θ$ , hence the same sequence attains the full worst-case risk.

5.1step 2.1step 3.1step 4.1algebra∎

Taking the supremum in [step 2.1], using the universal upper bound [step 3.1] and the attaining sequence [step 4.1], yields the exact equality and the stated zero-cutoff convention.

Remarks

This is expected squared loss in the Gaussian sequence experiment, over a weighted ellipsoid of coefficient sequences. It is not a deterministic worst-case error bound for arbitrary norm-bounded data perturbations.

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Sources

Litao Ding and Peter Mathé, Minimax rates for statistical inverse problems under general source conditions, Lemma 1 and its complete proof, PDF pp. 2–3 (standard reference, not scraped)
Iain M. Johnstone, Gaussian estimation: Sequence and wavelet models, §3.2, equations (3.11)–(3.12), printed pp. 60–61 (standard reference, not scraped)