Simultaneous fixed-target FPCR wild-bootstrap inference

Version 0.18 adds an explicit familywise calibration layer for a predeclared set of fixed Gaussian FPCR target trajectories.

It does not replace or alter the 0.16 heteroscedastic wild-bootstrap generator. Instead, it reuses the studentized roots already stored by wild_bootstrap_fpca_projection().

Scientific question

The base 0.16 method answers:

What is a target-wise confidence interval for the centered FPCR projection of each fixed target trajectory under heterogeneous response errors?

The 0.18 layer answers the stronger familywise question:

What common calibration protects the complete declared target family against at least one interval miss?

This is a fixed-target familywise statement. It is not a joint future-outcome prediction region.

Start from a qualified base result

base = wild_bootstrap_fpca_projection(
    trajectories,
    outcome,
    targets=declared_targets,
    n_bootstrap=1000,
    residual_components=k,
    inference_components=h,
    multiplier="normal",
    confidence_level=0.95,
    random_state=2028,
)

The base result already contains one studentized root for every bootstrap replicate and target.

Version 0.18 does not regenerate those roots.

Max-|t| post-calibration

Let T[b, j] denote the studentized bootstrap root for bootstrap replicate b and target j.

For each bootstrap replicate, compute

M[b] = max_j |T[b, j]|

The familywise critical value is the requested empirical quantile of M, using the same conservative method="higher" convention as the base target-wise calibration.

The simultaneous interval for target j is

reference_projection[j] ± critical_value * reference_se[j]

Because every target-wise absolute root is bounded by the replicate-wise maximum, the resulting simultaneous interval cannot be narrower than the corresponding same-level target-wise interval apart from floating-point tolerance.

For a family containing exactly one target, the simultaneous and target-wise calibrations are identical.

Declare the family before inspecting results

The family is exactly all target trajectories stored in the supplied base result.

To analyze a different family, construct a base result with that target set and then post-calibrate it.

Do not inspect target-wise intervals, remove inconvenient targets, and then describe the remaining family as if it had been predeclared. Changing the family changes the calibration target.

Why reuse the same bootstrap roots?

All targets must be evaluated under the same bootstrap replicate to preserve their empirical dependence structure.

Running a separate bootstrap for each target and combining marginal critical values afterward would discard that coupling.

The 0.18 implementation therefore:

• reuses the exact base studentized-root matrix;
• performs no new random-number generation;
• does not refit FPCA or score regressions;
• does not re-estimate residuals;
• does not rerun the wild bootstrap.

Target-wise versus simultaneous output

The result retains both:

• same-level target-wise critical values recomputed from each target's absolute roots;
• one familywise max-|t| critical value shared by all targets.

This makes the multiplicity cost visible rather than hiding it.

Interaction with truncation selection

The simultaneous layer is conditional on the k=g and h settings in the supplied base result.

If h was selected using data-driven stabilized-volatility rules, that selection uncertainty is not propagated automatically into the simultaneous interval.

For confirmatory work, define the target family and truncation strategy before inspecting final intervals.

Assumptions inherited from the base wild bootstrap

All 0.16 assumptions remain in force:

• functional regressors and FPCA/MFPCA score geometry are fixed during wild resampling;
• curve rows are independent sampling units;
• the scalar response model is Gaussian FPCR;
• residual/pseudo-truth truncation is k=g;
• inference uses explicit h>=g;
• multiplier choice is explicit;
• heteroscedastic studentization is recomputed inside each bootstrap pseudo-sample.

What simultaneity does and does not cover

The familywise statement covers only the fixed target projections included in the base result.

It does not provide simultaneous coverage for:

• future observed scalar outcomes;
• target trajectories not in the declared family;
• preprocessing choices;
• data-driven component-selection uncertainty;
• repeated-participant or clustered wild-bootstrap dependence;
• non-Gaussian/binomial functional regression.

Reporting checklist

Report:

1. the number and scientific definition of targets in the family;
2. the confidence level;
3. k, g=k, and h;
4. multiplier family and bootstrap replicate count;
5. the independent sampling unit;
6. that the FPCA basis/regressors were fixed;
7. that bootstrap-level heteroscedastic studentization was used;
8. the familywise max-|t| critical value;
9. that the same root matrix was reused without a second bootstrap;
10. that simultaneity is restricted to the declared fixed targets.

API links

• wild_bootstrap_fpca_projection()
• FPCAWildBootstrapProjectionResult
• FPCAWildBootstrapSimultaneousResult
• fpca_wild_bootstrap_projection_simultaneous_interval()
• fpca_wild_bootstrap_simultaneous_frame()
• plot_fpca_wild_bootstrap_simultaneous_interval()
• fpca_wild_bootstrap_simultaneous_reporting_text()

See also Heteroscedastic FPCR wild bootstrap, Stabilized-volatility FPCR selection, and References.

From simultaneous intervals to explicit tests

Version 0.19 uses the same declared fixed-target family and the same stored studentized root matrix to report target-wise bootstrap probabilities, single-step maxT-adjusted probabilities, and a complete-family global maximum-statistic test.

This is a separate evidence layer rather than a new bootstrap generator. It also makes the strong-FWER boundary explicit: the package does not assume subset pivotality or claim a closed/step-down procedure for arbitrary subsets of null hypotheses.

See Fixed-family FPCR wild-bootstrap hypothesis tests.