A study on statistical theory of density estimation via the Bernstein polynomial and its application

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In this study, we focus on the nonparametric methods via the Bernstein polynomial approximation to estimate a function such as the probability density (independent and identically distributed setting) or the spectral density (stationary process). We also aimed at developing methods using some asymmetric kernels. Especially, we have proposed (i) additive or multiplicative (hence, nonnegative) bias corrections to the Bernstein-based estimator and (ii) a class of modified Bessel kernel-based estimators. Their asymptotic properties (bias, variance, mean integrated squared error, and so on), including the asymptotic normality, were derived rigorously. Furthermore, we have re-examined (iii) the gamma kernel density estimator to get a better performance in terms of MISE. Several numerical studies based on both simulated and real data sets were presented to confirm our results in this study.