This thesis makes contributions to statistical inference for functional data, focusing specifically on two key topics. The first topic addresses reliability analysis and uncertainty quantification for functional data. The second topic consists of statistical power analysis and sample size estimation. These two topics are motivated by the statistical analysis of biomechanical data, as both provide a foundation for valid inference in this field.

Paper I concerns the uncertainty quantification of test-retest reliability. The intraclass correlation coefficient (ICC) for functional data can be expressed as an ICC curve and an integrated ICC. This paper evaluates nonparametric bootstrap confidence sets for both the ICC curve and its integrated version. Furthermore, it provides researchers with validated recommendations for reporting reliability uncertainty when measurements are functions.

The remaining three papers focus on statistical power and sample size estimation for functional data.

Paper II compares six widely used local inferential methods using the omnibus power definition in order to estimate sample sizes. This paper proposes a step-by-step framework of practical recommendations for researchers who intend to perform a priori sample size estimation for functional data analysis. Moreover, it introduces an interactive tool to perform power analysis and sample size estimation.

Paper III builds on the finding in Paper II that noise smoothness interacts in complex ways with the true effect characteristics. Here, the true effect refers to actual differences between population mean functions. This paper examines this interaction in detail for two inferential methods under two definitions of power, omnibus power and sensitivity. Its central finding is that the relationship between data characteristics and statistical power is not universal, but depends fundamentally on which definition of power is used and on the structure of the underlying effect. This key insight directly motivates Paper IV.

Paper IV proposes a new sensitivity-based definition of statistical power named Functional Power. This new definition is calculated relative to the specific region of the domain where rejection of the null hypothesis is expected based on practical relevance, rather than over the full true non-null region, which is often unrealistic as it might include negligible differences. This paper further explores how current definitions of statistical power differ and explains the reasoning behind choosing a specific definition to determine sample size.

Together, this thesis demonstrates how the choice of methodology affects analysis when measurements are functions. In addition, it connects these findings to biomechanics and allows researchers in this field to better understand and apply the concepts to their own work.