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The estimation of parameters in the frequency spectrum of a seasonallypersistent stationary stochastic process is addressed. For seasonal persistenceassociated with a pole in the spectrum located away from frequency zero, a newWhittle-type likelihood is developed that explicitly acknowledges the locationof the pole. This Whittle likelihood is a large sample approximation to thedistribution of the periodogram over a chosen grid of frequencies, andconstitutes an approximation to the time-domain likelihood of the data, via thelinear transformation of an inverse discrete Fourier transform combined with ademodulation. The new likelihood is straightforward to compute, and as will bedemonstrated has good, yet non-standard, properties. The asymptotic behaviourof the proposed likelihood estimators is studied; in particular,$N$-consistency of the estimator of the spectral pole location is established.Large finite sample and asymptotic distributions of the score and observedFisher information are given, and the corresponding distributions of themaximum likelihood estimators are deduced. A study of the small sampleproperties of the likelihood approximation is provided, and its superiorperformance to previously suggested methods is shown, as well as agreement withthe developed distributional approximations.

Non-Regular Likelihood Inference for Seasonally Persistent Processes