Local cross-validation for spectrum bandwidth choice (Q2703262)

The following nonparametric kernel spectral density estimator is considered: NEWLINE\[NEWLINE\hat f_M(\lambda_j)=\sigma_M^{-1}\sum_k K(M\lambda_k)I(\lambda_j-\lambda_k),NEWLINE\]NEWLINE where \(I\) is the periodogram, \(\lambda_k=2\pi k/N\), \(N\) is the number of observations, \(K\) is a spectral window, \(\sigma_M\) is a normalizing factor, and \(M\) is a bandwidth. It is proposed to select the bandwidth by a cross-validation procedure which minimizes the local integrated mean squared error: NEWLINE\[NEWLINEIMSE(\nu,M)=2\pi N^{-1}\sum_{j=1}^{N-1} W_m(\lambda_j-\nu)E[(\hat f_M(\lambda_j)-f(\lambda_j))/f(\lambda_j)]^2,NEWLINE\]NEWLINE where \(\nu\) is the frequency at which the true spectral density is estimated, and \(W_m\) is some kernel function. The authors construct a cross-validation functional which estimates the main part of \(IMSE(\nu,M)\) and prove its consistency as \(N\to\infty\). Results of simulations for autoregression models are presented.