Theory and Methods of Statistics

13.10.3 Estimation of the Spectral Density

Since I(w_j) is approximately distributed as (ξ_j/2)f(w_j), where ${\xi }_j\sim {\chi }_2^2$, and E[I(w_j)] is approximately equal to f(w_j) (noting that E[ξ_j] = 2). Thus I(w_j) is almost an unbiased estimator of f(w_j). However, it is not a consistent estimator since Var[I(w_j)] is approximately equal to Var[ξ_j/2]f(w_j)² = f(w_j)² and this variance does not converge to zero as the sample size $n\to \infty$. In order to construct a consistent estimator of the spectral density at any point w ∈ (0, 1/2), we may use a weighted average of I(w