Quantifying uncertainty using confidence regions is a central goal of statistical inference. Nevertheless, confidence band methodologies in functional data analysis are still underdeveloped compared to estimation and hypothesis testing. In this work, we present a new methodology for constructing joint confidence bands for functional parameter estimates. (1) it is computationally fast because it is not based on resampling, and (2) it is built under a fairness constraint of false positive rate balanced across partitions of the domain of bands, so it is typical It is not only a global interpretation, but also a new local interpretation, and (3) it does not require estimation of the full covariance function, so it can be used in the case of fragmentary functional data. Simulations show excellent finite-sample behavior of the band compared to existing alternatives. The practical use of our bands is demonstrated in two of his case studies on sports biomechanics and piecemeal growth curves.

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