A Bernstein polynomial estimation of the moment generating function

We study Bernstein polynomial smoothing for nonparametric estimation of the moment generating function (MGF) of a distribution supported on (Formula presented.). We prove strong consistency of the Bernstein-smoothed MGF estimator and show that, when the smoothing degree m grows sufficiently fast, it shares the same (Formula presented.) asymptotic normal distribution as the empirical MGF estimator, with variance (Formula presented.). A second-order bias–variance expansion reveals that Bernstein smoothing yields a bias of order (Formula presented.) and a variance-reduction term of order (Formula presented.). This leads to the optimal smoothing rate (Formula presented.) for pointwise mean squared error (MSE) minimization. Simulation results confirm that the Bernstein estimator is especially effective in challenging boundary-dominated settings, notably for U-shaped beta models and for larger positive t values. Overall, Bernstein polynomial smoothing provides a simple, theoretically justified, and practically effective tool for nonparametric MGF estimation on compact supports.