Non‐parametric Curve Estimation by Wavelet Thresholding with Locally Stationary Errors

An important aspect in the modelling of biological phenomena in living organisms, whether the measurements are of blood pressure, enzyme levels, biomechanical movements or heartbeats, etc., 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, 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 of curve estimation for functions belonging to a Besov class with locally stationary errors, we derive a near‐optimal rate for the ‐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 case 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 stronger adaptivity result is possible by the use of a particular location and level dependent threshold obtained by minimizing Stein's unbiased estimate of the risk. In this respect, our work generalizes previous results, which cover the situation of correlated, but stationary errors. A natural application of our approach is the estimation of the trend function of non‐stationary time series under the model of local stationarity. The method is illustrated on both an interesting simulated example and a biostatistical data‐set, measurements of sheep luteinizing hormone, which exhibits a clear non‐stationarity in its variance.