Wavelet threshold estimators for data with stationary correlated noise are constructed by the following prescription. First, form the discrete wavelet transform of the data points. Next, apply a level-dependent soft threshold to the individual coefficients, allowing the thresholds to depend on the level in the wavelet transform. Finally, transform back to obtain the estimate in the original domain. The threshold used at level j is s j p 2 log n, where s j is the standard deviation of the coefficients at that level, and n is the overall sample size. The minimax properties of the estimators are investigated by considering a general problem in multivariate normal decision theory, concerned with the estimation of the mean vector of a general multivariate normal distribution subject to squared error loss. An ideal risk is obtained by the use of an `oracle' that provides the optimum diagonal projection estimate. This `benchmark' risk can be considered in its own right as a measure of the s...

this paper so we will not mention this explicitly. The ideas and methods of Calderon and Zygmund [7] in harmonic analysis have shown that although we are not able to find the

In recent years there has been a considerable development in the use of wavelet methods in statistics. As a result, we are now at the stage where it is reasonable to consider such methods to be another standard tool of the applied statistician rather than a research novelty. With that in mind, this article is intended to give a relatively accessible introduction to standard wavelet analysis and to provide an up to date review of some common uses of wavelet methods in statistical applications. It is primarily orientated towards the general statistical audience who may be involved in analysing data where the use of wavelets might be e ective, rather than to researchers already familiar with the eld. Given that objective, we do not emphasise mathematical generality or rigour in our exposition of wavelets and we restrict our discussion to the more frequently employed wavelet methods in statistics. We provide extensive references where the ideas and concepts discussed can be followed up in...

This article defines and studies a new class of non-stationary random processes constructed from discrete non-decimated wavelets which generalizes the Cramer (Fourier) representation of stationary time series. We define an evolutionary wavelet spectrum (EWS) which quantifies how process power varies locally over time and scale. We show how the EWS may be rigorously estimated by a smoothed wavelet periodogram and how both these quantities may be inverted to provide an estimable time-localized autocovariance. We illustrate our theory with a pedagogical example based on discrete nondecimated Haar wavelets and also a real medical time series example.

We consider wavelet estimation of the time--dependent (evolutionary) power spectrum of a locally stationary time series. Hereby, wavelets are used to provide an adaptive local smoothing of a short--time periodogram in the time--frequency plane. For this, in contrast to classical nonparametric (linear) approaches we use non--linear thresholding of the empirical wavelet coe#cients. We show how these techniques allow for both adaptively reconstructing the local structure in the time--frequency plane and for denoising the resulting estimates. To this end a threshold choice is derived which results into a near--optimal L 2 --minimax rate for the resulting spectral estimator. Our approach is based on a 2--d orthogonal wavelet transform modified by using a cardinal Lagrange interpolation function on the finest scale. As an example, we apply our procedure to a time--varying spectrum motivated from mobile radio propagation. 1 Introduction Estimating power spectra which (slowly) change over ...

This article reviews the role of wavelets in statistical time series analysis. We survey work that emphasises scale such as estimation of variance and the scale exponent of a process with a specific scale behaviour such as 1/f processes. We present some of our own work on locally stationary wavelet (lsw) processes which model both stationary and some kinds of non-stationary processes. Analysis of time series assuming the lsw model permits identification of an evolutionary wavelet spectrum (ews) that quantifies the variation in a time series over a particular scale and at a particular time. We address estimation of the ews and show how our methodology reveals phenomena of interest in an infant electrocardiogram series.

Mallat, Papanicolaou and Zhang [MPZ98] recently proposed a method for approximating the covariance of a locally stationary process by a covariance which is diagonal in a specially constructed Coifman–Meyer basis of cosine packets. In this paper we extend this approach to estimating the covariance from sampled data. Our method combines both wavelet shrinkage and cosine-packet best-basis selection in a simple and natural way. The resulting algorithm is fast and automatic. The method has an interpretation as a nonlinear, adaptive form of anisotropic timefrequency smoothing. We introduce a new class of locally stationary processes which exhibits a form of inhomogeneous nonstationarity; our processes have covariances which typically change little from row to row, but might occasionally change abruptly. We study performance in an asymptotic setting involving triangular arrays of processes which are becoming increasingly stationary, and are able to prove rates of convergence results for our estimator. For this class of processes, the algorithm has advantages over traditional approaches like fixed-window-length segmentation followed by autocovariance estimation.

With this article we first like to a give a brief review on wavelet thresholding methods in non-Gaussian and non-i.i.d. situations, respectively. Many of these applications are based on Gaussian approximations of the empirical coefficients. For regression and density estimation with independent observations, we establish joint asymptotic normality of the empirical coefficients by means of strong approximations. Then we describe how one can prove asymptotic normality under mixing conditions on the observations by cumulant techniques. In the second part, we apply these non-linear adaptive shrinking schemes to spectral estimation problems for both a stationary and a non-stationary time series setup. For the latter one, in a model of Dahlhaus ([Da93]) on the evolutionary spectrum of a locally stationary time series, we present two different approaches. Moreover, we show that in classes of anisotropic function spaces an appropriately chosen wavelet basis automatically adapts to possibly dif...

In the modeling of biological phenomena, in living organisms whether the measurements are of blood pressure, enzyme levels, biomechanical movements or heartbeats, etc., one of the important aspects is time variation in the data. Thus, the recovery of a "smooth" regression or trend function from noisy time--varying sampled data becomes a problem of particular interest. Here we use non--linear wavelet thresholding to estimate a regression or a trend function in the presence of additive noise which, in contrast to most existing models, does not need to be stationary. (Here, nonstationarity means that the spectral behaviour of the noise is allowed to change slowly over time). We develop a procedure to adapt existing threshold rules to such situations, e.g., that of a time--varying variance in the errors. Moreover, in the model of curve estimation for functions belonging to a Besov class with locally stationary errors, we derive a near--optimal rate for the L 2 --risk between the unknown fu...

. We consider nonparametric estimation of the parameter functions a i (\Delta) , i = 1; : : : ; p , of a time-varying autoregressive process. Choosing an orthonormal wavelet basis representation of the functions a i , the empirical wavelet coefficients are derived from the time series data as the solution of a least squares minimization problem. In order to allow the a i to be functions of inhomogeneous regularity, we apply nonlinear thresholding to the empirical coefficients and obtain locally smoothed estimates of the a i . We show that the resulting estimators attain the usual minimax L 2 -rates up to a logarithmic factor, simultaneously in a large scale of Besov classes. The finite--sample behaviour of our procedure is demonstrated by application to two typical simulated examples. 1991 Mathematics Subject Classification. Primary 62M10; secondary 62F10 Key words and phrases. Nonstationary processes, time series, wavelet estimators, time-varying autoregression, nonlinear thresholdi...