Uses of cumulants in wavelet analysis

Cumulants are useful in studying nonlinear phenomena and in developing (approximate) statistical properties of quantities computed from random process data. Wavelet analysis is a powerful tool for the approximation and estimation of curves and surfaces. This work considers wavelets and cumulants, developing some sampling properties of wavelet fits to a signal in the presence of additive stationary noise via the calculus of cumulants. Of some concern is the construction of approximate confidence bounds around a fit. Both linear and shrunken wavelet estimates are considered. Extensions to spatial processes, irregularly observed processes and long memory processes are discussed. The usefulness of the cumulants lies in their employment to develop some of the statistical properties of the estimates.

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