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Supremum sieves admit tests in fixed-design Gaussian regression

Statement

Let $x_{1}, \dots, x_{n}$ be deterministic, $Y_{i} = w_{0} (x_{i}) + σ_{0} Z_{i}$ under the truth with independent $Z_{i} \sim N (0, 1)$ , and let candidate parameters be $(w, σ)$ with $σ \in [a, b]$ , $0 < a < b$ , and $σ_{0} \in [a, b]$ . Put $∥ u ∥_{n}^{2} = n^{- 1} \sum_{i} u (x_{i})^{2}$ . Suppose $ϵ_{n} \to 0$ , $n ϵ_{n}^{2} \to \infty$ , and a measurable function sieve $B_{n}$ has a finite supremum-norm cover of radius $3 ϵ_{n}$ with at most $\exp (6 C n ϵ_{n}^{2})$ centers, for a fixed $C > 0$ . For every $T > 0$ , there is a sufficiently large fixed $M$ and a measurable test $ϕ_{n}$ such that, for all sufficiently large $n$ , $P_{0}^{n} (ϕ_{n} = 1) \leq e^{- T n ϵ_{n}^{2}}, \sup_{\begin{array}{c} w \in B_{n}, σ \in [a, b] \\ ∥ w - w_{0} ∥_{n} + ∣ σ - σ_{0} ∣ > M ϵ_{n} \end{array}} P_{w, σ}^{n} (ϕ_{n} = 0) \leq e^{- T n ϵ_{n}^{2}} .$

Facts & Assumptions

Given: The deterministic design, true independent Gaussian errors, compact positive scale interval, rate sequence and finite measurable supremum cover.

[A1]

Assume AC The Axiom of Choice to select a finite cover for each n.

[F1]

Linear and quadratic Gaussian statistics have exponential tails: if $∥ c ∥_{n}$ is fixed, $n^{- 1} \sum_{i} c_{i} σ Z_{i}$ has tails at scale $∥ c ∥_{n} / \sqrt{n}$ ; if $∥ δ ∥_{n} \leq 1$ , the centered residual-energy statistic $n^{- 1} \sum_{i} (δ_{i} + σ Z_{i})^{2} - (σ^{2} + ∥ δ ∥_{n}^{2})$ has tails bounded by $2 e^{- n t^{2} / (4 V_{b})}$ for $t \leq b^{2} + 2$ .

Proof

Given: Fix a cover $v_{1}, \dots, v_{N_{n}}$ of $B_{n}$ with $∥ v_{j} - v ∥_{\infty} < 3 ϵ_{n}$ for the covered $v$ , and $N_{n} \leq e^{6 C n ϵ_{n}^{2}}$ .

1.1A1F1givenconstruct

For each center put $δ_{j i} = v_{j} (x_{i}) - w_{0} (x_{i})$ and $ρ_{j} = ∥ δ_{j} ∥_{n}$ . For centers with $ρ_{j} \geq M ϵ_{n} / 4$ , define $U_{j} = n^{- 1} \sum_{i} δ_{j i} (Y_{i} - w_{0} (x_{i}))$ and $ϕ_{j} = 1 {U_{j} \geq ρ_{j}^{2} / 4}$ . Under the truth, $U_{j}$ is centered normal with variance $σ_{0}^{2} ρ_{j}^{2} / n$ , so [F1] gives $P_{0}^{n} (ϕ_{j} = 1) \leq \exp [- n ρ_{j}^{2} / (32 b^{2})]$ .

1.2givenalgebra

Put $Δ = M ϵ_{n} / 2$ and $Q_{n} = n^{- 1} \sum_{i} (Y_{i} - w_{0} (x_{i}))^{2}$ . If $∥ w - w_{0} ∥_{n} \leq Δ$ and $∣ σ - σ_{0} ∣ > Δ$ , then under $(w, σ)$ , $E Q_{n} = σ^{2} + ∥ w - w_{0} ∥_{n}^{2}$ . For an upper scale alternative, this mean is at least $σ_{0}^{2} + a M ϵ_{n}$ ; for a lower scale alternative, it is at most $σ_{0}^{2} - a M ϵ_{n} + Δ^{2} \leq σ_{0}^{2} - a M ϵ_{n} / 2$ once $M ϵ_{n} \leq 2 a$ . Define $ϕ_{+} = 1 {Q_{n} \geq σ_{0}^{2} + a M ϵ_{n} / 4}$ and $ϕ_{-} = 1 {Q_{n} \leq σ_{0}^{2} - a M ϵ_{n} / 4}$ . Under the truth and on this scale-alternative region, the relevant deviations from the respective means are at least $a M ϵ_{n} / 4$ .

2.1F1step 1.1givenalgebra

If $∥ w - w_{0} ∥_{n} > M ϵ_{n} / 2$ , choose a cover center $v_{j}$ within $3 ϵ_{n}$ of $w$ . For $M \geq 12$ , $ρ_{j} \geq (M / 2 - 3) ϵ_{n} \geq M ϵ_{n} / 4$ ; for $M \geq 24$ , $ρ_{j} \geq 6 ϵ_{n}$ . Under $(w, σ)$ , $E U_{j} = ⟨ v_{j} - w_{0}, w - w_{0} ⟩_{n} \geq ρ_{j}^{2} - 3 ϵ_{n} ρ_{j} \geq ρ_{j}^{2} / 2$ . The threshold is therefore at least $ρ_{j}^{2} / 4$ below the mean, and [F1] gives $P_{w, σ}^{n} (ϕ_{j} = 0) \leq \exp [- n ρ_{j}^{2} / (32 b^{2})] \leq \exp [- M^{2} n ϵ_{n}^{2} / (512 b^{2})]$ .

3.1F1step 1.1step 2.1step 1.2givenchoose∎

For sufficiently large n, $Δ \leq 1$ and $a M ϵ_{n} / 4 \leq b^{2} + 2$ , so [F1] bounds each null tail and each uniform scale-alternative error by $2 \exp [- n a^{2} M^{2} ϵ_{n}^{2} / (64 V_{b})]$ . Let $J_{n} = {j : ρ_{j} \geq M ϵ_{n} / 4}$ be the finite set of centers retained in step 1.1, and set $ϕ_{n} = ϕ_{+} \lor ϕ_{-} \lor \max_{j \in J_{n}} ϕ_{j}$ , defining the last maximum to be $0$ when $J_{n} = \emptyset$ . By step 2.1 every mean-separated alternative has a center in $J_{n}$ ; all other alternatives are covered by the scale tests from step 1.2. The null error is at most $e^{6 C n ϵ_{n}^{2}} e^{- M^{2} n ϵ_{n}^{2} / (512 b^{2})} + 4 e^{- n a^{2} M^{2} ϵ_{n}^{2} / (64 V_{b})}$ . Choose fixed $M$ large enough that both bounds are at most $e^{- T n ϵ_{n}^{2}}$ for all sufficiently large n.

Source notes

The 2008 GP article states the fixed-design theorem but gives only a sketch of its testing step. The non-IID article supplies the average-Hellinger product setting. The explicit mean-score and residual-energy tests above provide the complete local argument for the claimed Gaussian model.

Depends on

Used by

Gaussian-process posterior contraction in density and regression models Theorem

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

A. W. van der Vaart and J. H. van Zanten, Rates of contraction of posterior distributions based on Gaussian process priors, §3.3 (standard reference, not scraped)
Subhashis Ghosal and Aad W. van der Vaart, Convergence rates of posterior distributions for noniid observations, §3 (standard reference, not scraped)