Alternative Estimators of Wavelet Variance

The wavelet variance is a scale-based decomposition of the process variance that is particularly well suited for analyzing intrinsically stationary processes. This decomposition has proven to be useful for studying various geophysical time series, including some related to sub-tidal sea level variations, vertical shear in the ocean and variations in soil composition along a transect. Previous work has established the large sample properties of an unbiased estimator of the wavelet variance formed using the non-boundary wavelet coefficients from the maximal overlap discrete wavelet transform (MODWT). The present work considers two alternative estimators, one of which is unbiased, and the other, biased. The new unbiased estimator is appropriate for asymmetric wavelet filters such as the Debaucheries filters of width four and higher and is obtained from the non-boundary coefficients that result from running a wavelet filter through a time series in both a forward and a backward direction. The biased estimator is constructed in a similar fashion, but utilizes all wavelet coefficients that result from filtering a time series in forward and backward directions. While the two alternative estimators have the same asymptotic distribution as the original unbiased estimator (with some restrictions in the case of the biased estimator), they can have substantially better statistical properties in small sample sizes. Formula for evaluating the mean squared errors of the usual unbiased estimator and the two alternative estimators are developed and verified for several fractionally differenced processes via Monte Carlo experiments