Spectral representation and estimation for locally

This article develops a wavelet decomposition of a stochastic process which parallels a time{localized Cram er (Fourier) spectral representation. We provide a time{scale instead of a time{frequency decomposition and, hence, instead of thinking as scale in terms of \inverse frequency" we start from genuine time{scale building blocks or \atoms". Using this class of locally stationary wavelet (LSW) processes, a doubly{indexed array of processes fX t;T g t=1;:::;T ; T 1, we develop a theory for the estimation of the \evolutionary wavelet spectrum". Our asymptotics are based on rescaling in time{ location which allows us to perform rigorous estimation theory starting from a single stretch of observations of fX t;T g. This evolutionary wavelet spectrum measures the local power in the variance{covariance decomposition of the process fX t;T g at a certain scale and a (rescaled) time location. It is possible to estimate the evolutionary wavelet spectrum by means of a wavelet periodogram or scalogram: in other words the squared coeecients from a discrete or stationary wavelet transform (SWT). However, in order to estimate the spectrum consistently we have to smooth the periodogram. In this article we choose to smooth by non{linear wavelet shrinkage of the wavelet periodogram treated as a function of rescaled time{location with respect to another orthogonal wavelet basis (this parallels recent work in curve estimation). Finally, we suggest an inverse transformation of the smoothed wavelet periodogram that estimates a localized autocovariance of the original stochastic process. Some numerical simulations and an application to a medical time series, which shows a particular non{stationary behaviour, indicate the usefulness of the SWT in our approach.