Separable Bands: Adaptive Nonparametric Inference

• Separable bands are confidence bands for nonparametric inference constructed using separable Gaussian processes with rigorous anti-concentration properties ensuring valid adaptive coverage.
• The methodology employs a Gaussian multiplier bootstrap and adaptive Lepski’s method to estimate quantiles without relying on classical extreme value distributions.
• Applications extend to density estimation, regression, and deconvolution, effectively balancing bias and variance through data-driven resolution selection.

Separable bands, in the context of adaptive confidence bands for nonparametric inference, are rigorously constructed objects whose theoretical foundation hinges on anti-concentration properties of Gaussian processes. This concept is central to the development of honest and adaptive uniform inference procedures, as exemplified in the work introducing the generalized Smirnov–Bickel–Rosenblatt (SBR) condition and multiplier bootstrap methodologies (Chernozhukov et al., 2013). The separability of the underlying Gaussian processes provides the crucial structure enabling the construction of bands with solid coverage guarantees, even in scenarios where classical limit distributions for the supremum are either unknown or fail to exist.

1. Generalized SBR Condition and Anti-concentration

Classical approaches to confidence bands for nonparametric densities rely on the SBR condition, demanding convergence to a known extreme value distribution (e.g., Gumbel) for the supremum of a studentized empirical process or its Gaussian analogue. The generalized SBR condition dispenses with such restrictive requirements. Formally, for a separable Gaussian process $G_{n,f}$ approximating the empirical process, the anti-concentration function

$$p_\varepsilon(Y) := \sup_{x\in\mathbb{R}} \Pr\left\{ \left| \sup_{t\in T} Y_t - x \right| \leq \varepsilon \right\}$$

must tend to zero, or quantitatively satisfy

$$\sup_{f\in\mathcal{F}} p_{\varepsilon_n}(|G_{n,f}|) \leq C_1\varepsilon_n \sqrt{\log n}$$

for sequences $\varepsilon_n \sqrt{\log n} \to 0$. Thus, the probability mass of the supremum does not concentrate too rapidly near any value, which is analytically justified for separable Gaussian processes. This property is foundational for building confidence bands in settings lacking explicit limit distributions.

2. Anti-concentration for Separable Gaussian Processes

The anti-concentration inequality is derived for separable Gaussian processes $(X_t)_{t \in T}$ with mean zero and unit variance. The Lévy concentration function of the supremum satisfies, for all $\varepsilon \geq 0$,

$p_\varepsilon(X) \leq 4 (a(X) + 1)$

where $a(X) = \mathbb{E}[\sup_{t\in T} X_t]$. Similar bounds hold for $\sup_{t \in T} |X_t|$. This result ensures that, even with large index sets and increasing complexity, the supremum spreads out its probability, a key technical ingredient underpinning the construction of separable bands with honest coverage. Importantly, such anti-concentration controls are valid regardless of explicit centering or scaling formulas for the supremum.

3. Separable Gaussian Process Approximations and Confidence Bands Construction

In nonparametric confidence band methodology, one first constructs a studentized process:

$$Z_{n,f}(x, l) = \frac{\sqrt{n} (\hat{f}_n(x, l) - f(x))}{\sigma_{n,f}(x, l)}$$

where $\hat{f}_n$ is a kernel density estimator and $\sigma_{n,f}$ its standard deviation. To avoid reliance on explicit limit laws for $\sup_{(x, l)} |Z_{n,f}(x, l)|$, the process is approximated by a separable Gaussian process $G_{n,f}$ matching the covariance structure of $Z_{n,f}$. The separability property ensures supremum calculations over $\mathcal{V}_n$ ($\mathcal{X} \times \mathcal{L}_n$) are justified, supporting both anti-concentration inequalities and coupling arguments essential for correct asymptotic coverage.

The critical value for the confidence band is defined as the quantile:

$$c_{n,f}(\alpha) = (1-\alpha)\text{-quantile of } \|\ G_{n,f} \|_{\mathcal{V}_n}$$

with separability guaranteeing that this quantile is well-defined in practice.

4. Gaussian Multiplier Bootstrap for Quantile Estimation

Because the analytic distribution of $\|\ Z_{n,f} \|_{\mathcal{V}_n}$ and $c_{n,f}(\alpha)$ is unknown in most practical cases, a Gaussian multiplier bootstrap is deployed:

• Independent normal multipliers $\xi_1, \dots, \xi_n$ are generated.
• The bootstrap process

$$\hat{\mathbb{G}}_n(x, l) = \frac{1}{\sqrt{n}} \sum_{i=1}^n \xi_i \frac{K_l(X_i, x) - \hat{f}_n(x, l)}{\hat{\sigma}_n(x, l)}$$

is computed, with $K_l$ the kernel function.

• The conditional $(1-\alpha)$ quantile $\hat{c}_n(\alpha)$ of the supremum

$$\left\| \hat{\mathbb{G}}_n \right\|_{\mathcal{V}_n} = \sup_{(x,l)\in\mathcal{V}_n} |\hat{\mathbb{G}}_n(x, l)|$$

is estimated over the observed data.

Confidence bands are constructed as

$$\mathcal{C}_n(x) = \left[\hat{f}_n(x, \hat{l}_n) - \frac{(\hat{c}_n(\alpha) + c_n') \hat{\sigma}_n(x, \hat{l}_n)}{\sqrt{n}}, \ \hat{f}_n(x, \hat{l}_n) + \frac{(\hat{c}_n(\alpha) + c_n')\hat{\sigma}_n(x, \hat{l}_n)}{\sqrt{n}} \right]$$

where $\hat{l}_n$ is adaptively determined, and $c_n'$ corrects for bias. This bootstrap-based quantile estimation exploits the separable structure to yield bands with valid coverage even in adaptive or data-driven scenarios.

5. Adaptive Resolution via Multiplier Bootstrap Lepski’s Method

A practical, data-driven adaptation strategy is implemented using a bootstrap version of Lepski’s method:

• For each $(x, l, l')$ triple, compute

$$\widetilde{\mathbb{G}}_n(x, l, l') = \frac{1}{\sqrt{n}} \sum_{i=1}^n \xi_i \frac{(K_l(X_i, x) - K_{l'}(X_i, x)) - (\hat{f}_n(x, l) - \hat{f}_n(x, l'))}{\hat{\sigma}_n(x, l, l')}$$

with $\hat{\sigma}_n(x, l, l')$ a suitable “variation” estimator (with truncation for stability).

• The selected smoothing parameter is defined as

$$\hat{l}_n = \inf\left\{ l \in \mathcal{L}_n : \sup_{l' \in \mathcal{L}_{n,l}} \sup_{x \in \mathcal{X}} \frac{\sqrt{n} |\hat{f}_n(x, l) - \hat{f}_n(x, l')|}{\hat{\sigma}_n(x, l, l')} \leq q \tilde{c}_n(\gamma_n)\right\}$$

where $\mathcal{L}_{n,l}$ collects all $l' < l$ and $q > 1$ is a tuning constant, while $\tilde{c}_n(\gamma_n)$ is the bootstrap quantile over the index set.

This adaptive selection reliably balances bias and variance for the confidence bands, avoiding conservative maximal inequalities and supporting nearly optimal bandwidths.

6. Scope, Limitations, and Applications

The separable bands methodology enumerates several strengths:

• The generalized SBR (anti-concentration) framework applies even when limit laws are unavailable or violated, such as in adaptive density estimation.
• Separability of the process is essential for coupling and quantile approximation, making the approach robust to changes in regularity, smoothing parameter choice, and data-driven resolution.
• The methods extend beyond density estimation, applicable to nonparametric regression, deconvolution, and qualitative hypothesis testing, provided VC-type function class properties are verified.
• The adaptive bootstrap Lepski’s method yields confidence bands with honest coverage and adapts to unknown smoothness.

Potential limitations are technical: complexity bounds for function classes, behavior of bias, and the need for careful calibration of bootstrap and threshold parameters per problem.

Summary

Separable bands in adaptive nonparametric inference denote uniform bands—defined through supremum statistics over separable Gaussian processes—whose theoretical validity rests on anti-concentration properties rather than classical extreme value convergence. The key constructions involve approximating the studentized process by a separable Gaussian process, generating bootstrap replicates for quantile estimation, and adaptively selecting resolution levels via data-driven procedures. These techniques yield confidence bands with uniform (honest) coverage and optimally adaptive width, applicable even in regimes of unknown limit law and complex, data-driven estimator tuning. The framework thus generalizes and strengthens the construction of nonparametric confidence bands beyond the reach of classical methods (Chernozhukov et al., 2013).