  A fundamental endeavor in exoplanetary research is to characterize the bulk
compositions of planets via measurements of their masses and radii. With future
sample sizes of hundreds of planets to come from TESS and PLATO, we develop a
statistical method that can flexibly yet robustly characterize these
compositions empirically, via the exoplanet M-R relation. Although the M-R
relation has been explored in many prior works, they mostly use a power-law
model, with assumptions that are not flexible enough to capture important
features in current and future M-R diagrams. To address these shortcomings, a
nonparametric approach is developed using a sequence of Bernstein polynomials.
We demonstrate the benefit of taking the nonparametric approach by benchmarking
our findings with previous work and showing that a power-law can only
reasonably describe the M-R relation of the smallest planets and that the
intrinsic scatter can change non-monotonically with different values of a
radius. We then apply this method to a larger dataset, consisting of all the
Kepler observations in the NASA Exoplanet Archive. Our nonparametric approach
provides a tool to estimate the M-R relation by incorporating heteroskedastic
measurement errors into the model. As more observations will be obtained in the
near future, this approach can be used with the provided R code to analyze a
larger dataset for a better understanding of the M-R relation.
