Bootstrap-based Adaptive Window Selection (BAWS)

• Bootstrap-based Adaptive Window Selection (BAWS) is a family of procedures that use bootstrap-calibrated quantiles to adaptively select model complexity or historical windows.
• It employs pairwise comparisons to identify the smallest acceptable estimator or the largest admissible look-back window, effectively balancing bias and variance.
• The method leverages wild and moving-block bootstrap techniques to provide finite-sample guarantees and near-optimal risk bounds in both regression and financial forecasting contexts.

Bootstrap-based Adaptive Window Selection (BAWS) denotes bootstrap-calibrated procedures for adaptive selection over an ordered collection of candidates when the appropriate degree of localization, smoothing, or historical look-back is unknown. In "Bootstrap tuning in ordered model selection" (Spokoiny et al., 2015), the procedure selects the smallest model which satisfies an acceptance rule based on comparison with all larger models. In "Adaptive Window Selection for Financial Risk Forecasting" (Li et al., 1 Mar 2026), BAWS adaptively determines the window size in a sequential manner by comparing realized scores against a data-dependent threshold, which is evaluate based on an idea of bootstrap. Across these formulations, the common structure is to test whether enlarging the effective window remains admissible, while replacing analytically fixed thresholds by bootstrap-calibrated ones.

1. Scope and problem formulations

BAWS appears in two technically distinct settings. The first is a linear signal-plus-noise model,

$Y = X\theta + \varepsilon,$

where $X$ is a known design, $\theta \in \mathbb R^p$ is unknown, and $\varepsilon$ is a zero-mean noise vector with unknown and possibly heteroscedastic covariance

$$\Sigma = \operatorname{diag}(\sigma_1^2,\dots,\sigma_n^2).$$

An ordered family of linear estimators

$$\{\tilde\theta_m\}_{m\in M}, \qquad \tilde\theta_m = A_m Y,$$

is given, with ordering in the sense of nondecreasing variance. Typical examples are local averages over a window of width $m$ in nonparametric regression on a grid and truncated singular-value decomposition with cutoff $m$ in linear inverse problems (Spokoiny et al., 2015).

The second setting is sequential forecasting for a stochastic process $\{X_t\}_{t\ge 1}$, where $X_t \sim \mathbb P_t$ is unknown. The target parameter is

$X$0

and estimation is performed on a recent look-back window of length $X$1 through

$X$2

The aim is to choose, at each time $X$3, the largest window $X$4 that is still homogeneous so as to balance bias and variance (Li et al., 1 Mar 2026).

Aspect	Ordered model selection	Financial risk forecasting
Data structure	$X$5	sequential observations $X$6
Candidates	ordered linear estimators $X$7	look-back windows $X$8
Selection target	smallest accepted $X$9	largest admissible $\theta \in \mathbb R^p$0

This juxtaposition shows that BAWS is not tied to a single statistical model. A plausible implication is that the term denotes a family of bootstrap-calibrated adaptive selection rules whose operational form depends on whether the ordered index represents estimator complexity or historical span.

2. Smallest-accepted BAWS in ordered model selection

In the ordered linear-estimator formulation, BAWS is based on pairwise tests comparing $\theta \in \mathbb R^p$1 to more complex estimators $\theta \in \mathbb R^p$2 for $\theta \in \mathbb R^p$3. The guiding principle is to select the smallest $\theta \in \mathbb R^p$4 that is not significantly worse than any larger $\theta \in \mathbb R^p$5. For any pair $\theta \in \mathbb R^p$6, the test statistic is

$\theta \in \mathbb R^p$7

A weighted norm may also be used:

$\theta \in \mathbb R^p$8

where $\theta \in \mathbb R^p$9 is a known positive-definite matrix (Spokoiny et al., 2015).

Let $\varepsilon$0 denote the $\varepsilon$1-quantile of the distribution of $\varepsilon$2 under the null hypothesis that the $\varepsilon$3- and $\varepsilon$4-estimators differ only by noise. The selector is then

$\varepsilon$5

Operationally, one starts at $\varepsilon$6 and increases $\varepsilon$7 until finding the first index for which no pairwise test with any larger $\varepsilon$8 is significant at level $\varepsilon$9. That index is $$\Sigma = \operatorname{diag}(\sigma_1^2,\dots,\sigma_n^2).$$0.

This formulation is described as a new "smallest accepted" approach motivated by Lepski's method and multiple testing theory. Its purpose is adaptive model choice in an ordered family without prior information about the variance structure of the noise, including the heteroscedastic case. The corresponding estimator is $$\Sigma = \operatorname{diag}(\sigma_1^2,\dots,\sigma_n^2).$$1, and the abstract states that the resulting theory applies equally to estimation of the whole parameter vector, some subvector or linear mapping, as well as the estimation of a linear functional.

3. Bootstrap calibration, propagation, and finite-sample guarantees

The ordered-model BAWS thresholds are calibrated by a wild bootstrap. An initial pilot fit is formed, for instance by the most complex estimator,

$$\Sigma = \operatorname{diag}(\sigma_1^2,\dots,\sigma_n^2).$$2

with residuals

$$\Sigma = \operatorname{diag}(\sigma_1^2,\dots,\sigma_n^2).$$3

and diagonal weight matrix

$$\Sigma = \operatorname{diag}(\sigma_1^2,\dots,\sigma_n^2).$$4

Given i.i.d. multiplier variables $$\Sigma = \operatorname{diag}(\sigma_1^2,\dots,\sigma_n^2).$$5 satisfying $$\Sigma = \operatorname{diag}(\sigma_1^2,\dots,\sigma_n^2).$$6 and $$\Sigma = \operatorname{diag}(\sigma_1^2,\dots,\sigma_n^2).$$7, bootstrap observations are generated as

$$\Sigma = \operatorname{diag}(\sigma_1^2,\dots,\sigma_n^2).$$8

and bootstrap estimators are

$$\Sigma = \operatorname{diag}(\sigma_1^2,\dots,\sigma_n^2).$$9

For each pair $$\{\tilde\theta_m\}_{m\in M}, \qquad \tilde\theta_m = A_m Y,$$0,

$$\{\tilde\theta_m\}_{m\in M}, \qquad \tilde\theta_m = A_m Y,$$1

and $$\{\tilde\theta_m\}_{m\in M}, \qquad \tilde\theta_m = A_m Y,$$2 is estimated by the empirical $$\{\tilde\theta_m\}_{m\in M}, \qquad \tilde\theta_m = A_m Y,$$3-quantile of the bootstrap replicates (Spokoiny et al., 2015).

The calibration is organized through the propagation condition:

$$\{\tilde\theta_m\}_{m\in M}, \qquad \tilde\theta_m = A_m Y,$$4

for each fixed $$\{\tilde\theta_m\}_{m\in M}, \qquad \tilde\theta_m = A_m Y,$$5. Under mild moment and design assumptions, including bounded operator norms of $$\{\tilde\theta_m\}_{m\in M}, \qquad \tilde\theta_m = A_m Y,$$6 and bounded leverage, the wild bootstrap scheme yields simultaneous control for all $$\{\tilde\theta_m\}_{m\in M}, \qquad \tilde\theta_m = A_m Y,$$7 up to a small remainder term of order $$\{\tilde\theta_m\}_{m\in M}, \qquad \tilde\theta_m = A_m Y,$$8.

The stated finite-sample bootstrap validity theorem gives constants $$\{\tilde\theta_m\}_{m\in M}, \qquad \tilde\theta_m = A_m Y,$$9, depending only on the design $m$0 and multiplier law, such that uniformly over all $m$1 and $m$2,

$m$3

Hence, with probability at least $m$4, the entire acceptance region

$m$5

is valid.

An oracle index $m$6 is defined by

$m$7

where

$m$8

The BAWS-selected estimator satisfies the oracle-type risk bound

$m$9

with $m$0 an absolute constant and

$m$1

Choosing $m$2 decreasing slowly with $m$3, such as $m$4, drives $m$5. The paper describes this as yielding nearly minimax-optimal adaptation over the ordered family.

4. Sequential BAWS for adaptive look-back selection

In the financial forecasting formulation, BAWS operates online. Once a window $m$6 is selected and $m$7 computed, the one-step-ahead realized score is

$m$8

Admissibility of a candidate window $m$9 is assessed by comparing its empirical loss against that of smaller windows. For each $\{X_t\}_{t\ge 1}$0 in $\{X_t\}_{t\ge 1}$1,

$\{X_t\}_{t\ge 1}$2

If $\{X_t\}_{t\ge 1}$3 is small for all $\{X_t\}_{t\ge 1}$4, enlarging the window to $\{X_t\}_{t\ge 1}$5 does not appreciably increase bias, and $\{X_t\}_{t\ge 1}$6 is deemed admissible (Li et al., 1 Mar 2026).

Given thresholds $\{X_t\}_{t\ge 1}$7, the pairwise decision rule is

$\{X_t\}_{t\ge 1}$8

A candidate $\{X_t\}_{t\ge 1}$9 is admissible, $X_t \sim \mathbb P_t$0, if $X_t \sim \mathbb P_t$1 for all $X_t \sim \mathbb P_t$2; otherwise $X_t \sim \mathbb P_t$3. The selected window is

$X_t \sim \mathbb P_t$4

This version therefore reverses the monotonic direction of the ordered-model selector. Instead of seeking the smallest acceptable complexity, it seeks the largest admissible history. That distinction is structural rather than cosmetic: in the forecasting setting, long windows reduce variance but may span structural changes, while short windows are more local but statistically noisier.

5. Bootstrap thresholds, change-point behavior, and risk-measure specialization

Thresholds in the sequential BAWS procedure are calibrated by bootstrap at each time $X_t \sim \mathbb P_t$5 and window $X_t \sim \mathbb P_t$6. With nominal acceptance level $X_t \sim \mathbb P_t$7, one generates $X_t \sim \mathbb P_t$8 bootstrap samples of size $X_t \sim \mathbb P_t$9; if the data are i.i.d. in $X$00, sampling is with replacement, while dependent data use the moving-block bootstrap with block length $X$01. For each bootstrap sample,

$X$02

and the replicate statistic at $X$03 is

$X$04

The threshold is the empirical $X$05-quantile,

$X$06

Under the pairwise null $X$07 that the blocks $X$08 and $X$09 are identically distributed,

$X$10

If strong family-wise error rate control at level $X$11 is desired, Bonferroni can be applied by replacing $X$12 with

$X$13

For the squared-loss case $X$14, with a single change-point inside a large window $X$15, the theorem labeled "Reject Overlong Windows" states that

$X$16

Thus BAWS rejects, with high probability, any window $X$17 spanning the unknown break at $X$18. A corollary for Gaussian blocks gives

$X$19

with

$X$20

The same framework is specialized to elicitable risk measures. For $X$21, the score is

$X$22

with

$X$23

For joint $X$24, the score is

$X$25

with suitable $X$26, and

$X$27

BAWS then runs unchanged with $X$28 replaced by the relevant consistent scoring function. This yields an online VaR or ES forecast that adapts the look-back to shifts in the tail distribution.

6. Algorithmic implementation, empirical behavior, and relation to adjacent methods

In the ordered-model version, the practical algorithm takes as input data $X$29, design $X$30, linear estimators $X$31, tuning $X$32, and bootstrap replications $X$33. The procedure computes the pilot fit, generates bootstrap replicates, estimates all $X$34, scans upward from $X$35, and outputs $X$36. The stated computational complexity is $X$37 to compute all bootstrap estimates and $X$38 comparisons. In practice one often uses $X$39–$X$40, $X$41–$X$42, and $X$43 (Spokoiny et al., 2015).

In the sequential forecasting version, the pseudo-code assumes a minimum window $X$44, a sparse candidate set $X$45, threshold level $X$46, and bootstrap replication count $X$47. Suggested candidate-window construction begins at $X$48, for example $X$49–$X$50, then uses a coarse grid with increments of $X$51 up to $X$52, $X$53 up to $X$54, $X$55 up to $X$56, $X$57 up to $X$58, and $X$59 thereafter, while dynamically including $X$60 and exploring upward from there in steps of $X$61. Typical $X$62 values are $X$63–$X$64, bootstrap replications $X$65 are $X$66–$X$67, and for dependent data one chooses $X$68 with $X$69. The computational cost is $X$70 model fits per $X$71, plus $X$72, with suggested accelerations including warm-start iterates, cached partial sums, and reduced $X$73 for large $X$74 (Li et al., 1 Mar 2026).

The comparative positioning of BAWS is explicit in both formulations. In ordered model selection, classical Lepski's method fixes thresholds by analytical bounding of $X$75 and requires knowledge or consistent estimation of $X$76, whereas BAWS replaces those fixed bounds by data-driven bootstrap quantiles and hence adapts automatically to heteroscedasticity or unknown noise structures. Against cross-validation, BAWS is described as enjoying nonasymptotic oracle bounds valid for finite $X$77 and controlling Type I errors in a multiple-testing framework. Numerical experiments in Section 5 show that BAWS matches or outperforms Lepski's method and generalized cross-validation in both homoscedastic and heteroscedastic settings, especially when the noise variance is spatially varying (Spokoiny et al., 2015).

In financial risk forecasting, BAWS is compared against SAWS, fixed rolling windows $X$78, $X$79, $X$80, and the full recursive window. In discrete mean/variance break scenarios, the metrics are mean absolute bias, variance, MSE, cumulative risk, and cumulative forecast loss. For Setting A1, the VaR forecast MSE values are

$X$81

and BAWS achieves the lowest MSE, CR and CL in all three discrete-break settings. In the GARCH$X$82 scenario with skewed-$X$83 innovations, using moving-block bootstrap with $X$84 and block length $X$85, the reported VaR MSE values are

$X$86

with BAWS again yielding the lowest MSE, CR, and CL. For S&P 500 daily losses over $X$87–$X$88, average forecast loss in $X$89–$X$90 is reported as $X$91 for BAWS, $X$92 for SAWS, $X$93 for $X$94, $X$95 for $X$96, $X$97 for $X$98, and $X$99 for the full window; for the GFC subperiod, the corresponding values are $\theta \in \mathbb R^p$00, $\theta \in \mathbb R^p$01, $\theta \in \mathbb R^p$02, $\theta \in \mathbb R^p$03, $\theta \in \mathbb R^p$04, and $\theta \in \mathbb R^p$05 (Li et al., 1 Mar 2026).

A recurrent misconception would be to treat BAWS as a single fixed rule with a single monotonic selection direction. The record presented in these two papers indicates otherwise: one formulation chooses the smallest accepted model in an ordered estimator family, while the other chooses the largest admissible look-back window in a sequential forecasting problem. What remains invariant is the bootstrap-calibrated acceptance logic under ordered comparisons.