M-Estimation of Wavelet Variance

The wavelet variance provides a scale-based decomposition of the process variance for a time series or random field. It has seen increasing use in geophysics, astronomy, genetics, hydrology, medical imaging, oceanography, soil science, signal processing and texture analysis. In practice, however, data collected in the form of a time series or random field often suffer from various types of contamination. We discuss the difficulties and limitations of existing contamination models (pure replacement models, additive outliers, level shift models and innovation outliers that hide themselves in the original time series) for robust nonparametric estimates of second-order statistics. We then introduce a new model based upon the idea of scale-based multiplicative contamination. This model supposes that contamination can occur and affect data at certain scales and thus arises naturally in multi-scale processes and in the wavelet variance context. For this new contamination model, we develop a full M -estimation theory for the wavelet variance and derive its large sample theory when the underlying time series or random field is Gaussian. Our approach treats the wavelet variance as a scale parameter and offers protection against contamination that operates additively on the log of squared wavelet coefficients and acts independently at different scales. Our asymptotic results rely on Hermite expansions and on the square integrability of the spectral density function of the wavelet coefficients. Under this setting, proofs of our theoretical results require extension and modification of the work of Koul and Surgailis (1997) and are intricate in its use of the famous Dahling–Taqqu chaining argument. Based upon computer experiments and a real data concerning pockets of open cells in the atmosphere, we find that our methods perform better than conventional methods for dealing with a contaminated time series or random field.