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Spectral analysis is a powerful, flexible technique for analysing the structure of a stochastic process. However, its application is impeded by the difficulty of choosing the appropriate smoothing to apply to the estimator of the spectral density function. A general framework is proposed which allows the construction of estimators quadratic in the data which incorporate any combination of smoothing methods. The combined effect of the smoothing may be characterised by a generalised smoothing function. The statistical properties of such estimators are then expressed in terms of the generalised smoothing function. It is shown that the bias of an estimator depends only on an equivalent lag or covariance weighting function, which can be easily calculated from the generalised smoothing function. After considering the types of smoothing which are or could be applied to estimators of the spectral density comparisons of some widely used procedures are made which illustrate how the generalised framework leads to a new method of comparing smoothing procedures. The examples suggest that tapering of data samples is less effective than the application of the equivalent lag window to the covariance estimator. The equivalent window generates the same bias, but has better variance performance. Finally, some asymptotic properties are established for estimators for Gaussian processes subjected to non-linear instantaneous transformations. Recent work which has established comprehensive conditions for the existence of an estimator is combined with new extensions to previous asymptotic results for certain transformations to give the asymptotic moments and distribution for a wide class of transformations. |
