Module bootstrap 

Bootstrap and robust control limit estimation.

Provides alternative methods for computing T-squared and SPE control limits when the parametric chi-squared assumption may not hold.

Note on bootstrap intervals

The bootstrap methods here use the percentile method. For small samples (n < 30), bias-corrected and accelerated (BCa) intervals may give better coverage, but are not yet implemented.

Convergence properties

The bootstrap percentile method has convergence rate O(n^{-1/2}) for quantile estimation. The BCa method (not yet implemented) achieves O(n^{-1}) via bias and skewness corrections (Efron & Tibshirani, 1993, Section 14.3).

The KDE estimator with Silverman bandwidth converges at rate O(n^{-4/5}) for smooth densities (the minimax-optimal rate for twice-differentiable densities). The bisection root-finding achieves machine-precision (~1e-10 relative) within 200 iterations.

Method selection guide

1. Parametric: use when SPE/T² is well-approximated by chi-squared (check via spe_moment_match_diagnostic).
2. Empirical: use for n > 50 when no distributional assumption is desired.
3. Bootstrap: use for n = 10–50 when parametric may be inaccurate.
4. KDE: use for smooth unimodal distributions with n > 30.

References

• Efron, B. & Tibshirani, R.J. (1993). An Introduction to the Bootstrap. Chapman & Hall/CRC. Section 13.3, pp. 178–182 (bootstrap methodology); Section 14.3 (BCa convergence).
• Silverman, B.W. (1986). Density Estimation for Statistics and Data Analysis. Chapman & Hall. Section 3.4.1, pp. 47–48 (kernel density estimation bandwidth selection).

Enums

ControlLimitMethod

Method for computing control limits.

Functions

spe_limit_robust

Compute a robust SPE control limit.

t2_limit_robust

Compute a robust T-squared control limit.