The wavelet periodogram is hard to smooth because of the low signal-to-noise ratio and non-stationary covariance structure. This article introduces a method for smoothing a local wavelet periodogram by applying a Haar-Fisz transform which approximately Gaussianizes and approximately stabilizes the variance of the periodogram. Consequently, smoothing the transformed periodogram can take advantage of the wide variety of existing techniques suitable for homogeneous Gaussian data. This article demonstrates the superiority of the new method over existing methods and supplies theory that proves the Gaussianizing, variance stabilizing and decorrelation properties of the Haar-Fisz transform
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