On estimation of the wavelet variance

The wavelet variance decomposes the variance of a time series into components associated with different scales. We consider two estimators of the wavelet variance: the first based upon the discrete wavelet transform, and the second, called the maximal-overlap estimator, based upon a filtering interpretation of wavelets. We determine the large sample distribution for both estimators and show that the maximal-overlap estimator is more efficient for a class of processes of interest in the physical sciences. We discuss methods for determining an approximate confidence interval for the wavelet variance. We demonstrate through Monte Carlo experiments that the large sample distribution for the maximal-overlap estimator is a reasonable approximation even for the moderate sample size of 128 observations. We apply our proposed methodology to a series of observations related to vertical shear in the ocean.