  Studies in circadian biology often use trigonometric regression to model
phenomena over time. Ideally, protocols in these studies would collect samples
at evenly distributed and equally spaced time points over a 24 hour period.
This sample collection protocol is known as an equispaced design, which is
considered the optimal experimental design for trigonometric regression under
multiple statistical criteria. However, implementing equispaced designs in
studies involving individuals is logistically challenging, and failure to
employ an equispaced design could cause a loss of statistical power when
performing hypothesis tests with an estimated model. This paper is motivated by
the potential loss of statistical power during hypothesis testing, and
considers a weighted trigonometric regression as a remedy. Specifically, the
weights for this regression are the normalized reciprocals of estimates derived
from a kernel density estimator for sample collection time, which inflates the
weight of samples collected at underrepresented time points. A search procedure
is also introduced to identify the concentration hyperparameter for kernel
density estimation that maximizes the Hessian of weighted squared loss, which
relates to both maximizing the $D$-optimality criterion from experimental
design literature and minimizing the generalized variance. Simulation studies
consistently demonstrate that this weighted regression mitigates variability in
inferences produced by an estimated model. Illustrations with three real
circadian biology data sets further indicate that this weighted regression
consistently yields larger test statistics than its unweighted counterpart for
first-order trigonometric regression, or cosinor regression, which is prevalent
in circadian biology studies.
