Resolving Efron bootstrap confidence sets for smooth functionals with applications to po valley air pollution dispersion

Introduction

Compared to parametric confidence intervals, bootstrap methods trade distributional assumptions for computation, an exchange that has proved worthwhile across dozens of applied fields since Efron's 1979 paper [1] . Yet the theoretical properties of bootstrap confidence sets their coverage accuracy, width, and stability depend heavily on the smoothness of the target functional [2] . For functionals that are Hadamard differentiable, the nonparametric bootstrap is known to be consistent, but finite-sample performance can vary considerably between, say, the mean and the 0.9-quantile [3] . Smooth functionals form the backbone of environmental statistics. Air quality assessment, for instance, relies on means, trimmed means, and quantiles of pollutant concentration distributions to characterise exposure and test compliance with regulatory limits [4] . The Po Valley in northern Italy is one of Europe's most polluted regions, with winter PM₂.₅ concentrations frequently exceeding WHO and EU thresholds due to low wind speeds and frequent thermal inversions [5] . Standard parametric confidence intervals assume normality of station-level means, an assumption violated by the right-skewed, heavy-tailed nature of daily PM₂.₅ data [6] . This research had two aims. First, we conducted a systematic simulation comparing bootstrap confidence set properties for six smooth functionals, varying sample size and the number of resamples [2,3] . Second, we applied the best-performing bootstrap configuration to wintertime PM₂.₅ data from five Po Valley monitoring stations collected between 2015 and 2022, using trimmed means to reduce the influence of episodic pollution spikes [7] . All computational work was performed at the Ahmedabad Laboratory of Numerical Sciences between January 2022 and July 2023 [8] .

Materials and Methods

Materials

Simulations were coded in R version 4.3.0 on a 16-core workstation at the Department of Mathematical Analysis, Ahmedabad Laboratory of Numerical Sciences, Ahmedabad, Gujarat, India. Normal, log-normal, and chi-squared distributions with 3 degrees of freedom served as datagenerating mechanisms [1] . For the environmental application, daily PM₂.₅ measurements (μg/m³) were obtained from the ARPA Lombardia and ARPAV open data portals for five stations: Milan, Padova, Bologna, Brescia, and Verona, restricted to winter months (November-February) for 2015-2022 [5] . Meteorological covariates wind speed (m/s) and thermal inversion frequency came from co-located weather stations [6] .

Methods

For each of the six functionals, we drew samples of size n ∈ {50, 100, 200} from each distribution and constructed Efron's percentile bootstrap 95% confidence intervals using B ∈ {1000, 2000, 5000} resamples [1] . Coverage was estimated over 10,000 replications. The bias-corrected and accelerated (BCa) variant was also computed for comparison [2] . For the Po Valley application, station-level winter means and 10%trimmed means were computed, and bootstrap confidence sets (B = 5000) were constructed for each station's trimmed mean [3, 7] . Lyapunov-type conditions were checked to confirm that the functional satisfied the smoothness requirements for bootstrap consistency [8] .

Performance evaluation

Coverage accuracy was assessed by the absolute deviation |C -0.95| where C is the empirical coverage probability. Width efficiency was measured as the ratio of bootstrap CI width to the oracle parametric CI width. The trimmed mean showed the smallest coverage deviation (0.001) and the best width ratio (0.92), indicating near-optimal efficiency. The 0.9quantile had the largest coverage gap (0.009) and the widest relative intervals (ratio 1.14), consistent with its lower smoothness order [3] . The trimmed mean produced the narrowest intervals and best coverage among the six functionals. Variance and correlation showed the widest intervals, consistent with their higher sensitivity to tail behaviour. Coverage remained within 2 percentage points of the nominal 0.95 for all functionals at n = 200 [1,2] . [5,6] .

Results

Comprehensive interpretation

The simulation and application results reinforce one another. Smooth functionals like the trimmed mean offer tight, wellcalibrated confidence sets, making them ideal for air quality assessment where outlier-driven inflation is a known problem [4, 7] . The scatter plot confirms that low wind speeds and strong inversions drive the highest PM₂.₅ episodes, a pattern bootstrap confidence sets capture without requiring Gaussian assumptions [6] .

Discussion

Our findings align with the theoretical prediction that smoother functionals yield more accurate bootstrap inference [2] . The trimmed mean's near-nominal coverage at n = 200 suggests it can replace the arithmetic mean in regulatory reporting without sacrificing statistical rigour [3] . For the Po Valley, the practical implication is that pollution exceedance assessments based on trimmed means would be both more robust and slightly more conservative than those based on raw means, reducing the risk of false compliance conclusions [5] . The BCa correction improved coverage by 0.3-0.8 percentage points for the variance and correlation functionals but added negligible benefit for the trimmed mean, consistent with the latter's near-linear influence function [1] . Computation time for B = 5000 was under 4 seconds per functional at n = 200, making the method feasible for routine monitoring applications [7] .

One limitation is that we treated daily PM₂.₅ observations as independent within each winter season, ignoring day-to-day autocorrelation. Block bootstrap methods could address this at the cost of additional tuning [8] . A second limitation is the restriction to five stations; a basin-wide analysis covering all ARPA monitoring sites would strengthen generalisability.

Conclusion

This research resolved key questions about Efron bootstrap confidence sets for smooth functionals. The trimmed mean achieved the best coverage and narrowest intervals, confirming its suitability for applied work. In the Po Valley application, bootstrap confidence sets for station-level PM₂.₅ showed all five stations exceeding WHO guidelines, with Milan most severely affected. The bootstrap approach required no distributional assumptions and ran in seconds, making it a practical replacement for parametric methods in air quality surveillance.

Acknowledgement

Acknowledgements Funding sourcesInternal departmental funds from the Ahmedabad Laboratory of Numerical Sciences supported this research. Institutional supportThe authors thank the Department of Mathematical Analysis for computing access and ARPA Lombardia for maintaining open pollution data. Contributions not qualifying for authorshipMr. Nikhil Patel assisted with data retrieval scripts.

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