A study on statistical theory of choosing a smoothing parameter for spectral density estimation and its application

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The smoothing parameter in nonparametric function estimation plays a role in controlling the trade off between the approximation error (in terminology of Statistics, it is nothing but bias of the estimator) and variance of the estimator. In this research, I discussed a new spectral density estimator/probability density estimator based on a generalized Bernstein polynomial approximation theory, in which the degree of polynomial and the additional second parameter are the smoothing parameters. Especially, main focus is the selection problem of these two parameters, where the latter second parameter enables us to generalize the classical Bernstein method. I established several formulas of asymptotic bias, variance and mean squared error as well as the asymptotic normality of the proposal.