Degree selection methods for curve estimation via Bernstein polynomials

Bernstein Polynomial (BP) bases can uniformly approximate any continuous function based on observed noisy samples. However, a persistent challenge is the data-driven selection of a suitable degree for the BPs. In the absence of noise, asymptotic theory suggests that a larger degree leads to better approximation. However, in the presence of noise, which reduces bias, a larger degree also results in larger variances due to high-dimensional parameter estimation. Thus, a balance in the classic bias-variance trade-off is essential. The main objective of this work is to determine the minimum possible degree of the approximating BPs using probabilistic methods that are robust to various shapes of an unknown continuous function. Beyond offering theoretical guidance, the paper includes numerical illustrations to address the issue of determining a suitable degree for BPs in approximating arbitrary continuous functions.