Non-parametric curve estimation by wavelet thresholding with locally stationary errors

An important aspect in the modelling of biological phenomena in living orga nisms, whether the measurements are of blood pressure, enzyme levels, biome chanical movements or heartbeats, etc., is time variation in the data. Thus , the recovery of a "smooth" regression or trend function from noisy time-v arying sampled data becomes a problem of particular interest. Here we use n on-linear wavelet thresholding to estimate a regression or a trend function in the presence of additive noise which, in contrast to most existing mode ls, does not need to be stationary. (Here, non-stationarity 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 o f curve estimation for functions belonging to a Besov class with locally st ationary errors, we derive a near-optimal rate for the L-2-risk between the unknown function and our soft or hard threshold estimator, which holds in the general case of an error distribution with bounded cumulants. In the ca se of Gaussian errors, a lower bound on the asymptotic minimax rate in the wavelet coefficient domain is also obtained. Also it is argued that a stron ger adaptivity result is possible by the use of a particular location and l evel dependent threshold obtained by minimizing Stein's unbiased estimate o f the risk. In this respect, our work generalizes previous results, which c over the situation of correlated, but stationary errors. A natural applicat ion of our approach is the estimation of the trend function of non-stationa ry time series under the model of local stationarity. The method is illustr ated on both an interesting simulated example and a biostatistical data-set , measurements of sheep luteinizing hormone, which exhibits a clear non-sta tionarity in its variance.