Bernstein polynomial estimation of a spectral density

We consider an application of Bernstein polynomials for estimating a spectral density of a stationary process. The resulting estimator can be interpreted as a convex combination of the (Daniell) kernel spectral density estimators at m points, the coefficients of which are probabilities of the binomial distribution bin( m − 1, | λ |/ π ), λ ∈ Π ≡ [− π , π ] being the frequency where the spectral density estimation is made. Several asymptotic properties are investigated under conditions of the degree m . We also discuss methods of data‐driven choice of the degree m . For a comparison with the ordinary kernel method, a Monte Carlo simulation illustrates our methodology and examines its performance in small sample.