University of Washington - Department of Statistics
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 subtidal 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 nonboundary 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 Daubechies filters of width four and higher. This estimator is based upon running the filter through the time series in both a forward and a backward direction. The biased estimator uses all available MODWT wavelet coefficients formed in conjunction with reflection boundary conditions. While the alternative estimators have the same asymptotic distribution as the original unbiased estimator, they can have substantially better statistical properties in small sample sizes. As an example, the original and alternative estimators are applied to estimate the wavelet variance of marine atmospheric boundary layer turbulence.