  Accelerometers enable an objective measurement of physical activity levels
among groups of individuals in free-living environments, providing
high-resolution detail about physical activity changes at different time
scales. Current approaches used in the literature for analyzing such data
typically employ summary measures such as total inactivity time or
compositional metrics. However, at the conceptual level, these methods have the
potential disadvantage of discarding important information from recorded data
when calculating these summaries and metrics since these typically depend on
cut-offs related to intensity exercise zones that are chosen subjectively or
even arbitrarily. Much of the data collected in these studies follow complex
survey designs, making application of standard statistical tools such as
non-parametric regression models inappropriate and the requirement of specific
estimation procedures according to particular sampling-design is mandatory.
With functional data or other complex objects, barely literature exist that
handles complex sampling designs in the statistical analysis. This paper aims
two-fold; first, we introduce a new functional representation of accelerometer
data of a distributional nature to build a complete individualized profile of
each subject's physical activity levels. Second, using the NHANES accelerometer
data (2003-2006), we show the potential advantages of this new representation
to predict patients' outcomes over $68$ years of age. A critical component in
our statistical modeling is that we extend non-parametric functional models
used: kernel smoother and kernel ridge regression, to handle the specific
effect of complex sampling design in order to provide reliable conclusions
about the influence of physical activity in distinct analysis performed.
