Statistical Properties for Coherence Estimators From Evolutionary Spectra

Evolutionary spectra were developed by Priestley to extend spectral analysis to some nonstationary time series, in particular semistationary processes, of which the ubiquitous uniformly modulated processes are a subclass. Coherence is well defined for bivariate semistationary processes and can be estimated from such processes. We consider Priestley's estimator for the evolutionary spectral density matrix, and show that its elements can be written as weighted multitaper estimators with calculable weights and tapers. Under Gaussianity an approximating Wishart-distribution model follows for the spectral matrix, valid for all frequencies except small computable intervals near zero and Nyquist. Moreover, the critically important degrees of freedom are known. Consequently, the statistical distribution of the coherence is given by Goodman's distribution and the raw coherence estimate can be accurately debiased. Theoretical results are verified using a model for wind fluctuations: Simulations give excellent agreement between the mean debiased coherence estimates and true coherence, and between the proposed and empirical distributions of coherence.