Wavelets, Longmemoryand Bootstrap - an Approach to Detecting Discontinuities

Weakly stationary time series are defined by a slow decay of autocorrelations and a pole of the spectral density at the origin. Sample paths tend to exhibit local spurious trends and cycles of varying length and magnitude. Visual separation of stationary long-memory components and deterministic trend functions is difficult. Statistical methods such as kernel and local polynomial smoothing and wavelet thresholding have been developed however in the literature for this purpose (see e.g. Hall and Hart 1990, Csörgö and Mielniczuk 1995, Ray and Tsay 1997, Robinson 1997, Beran and Feng 2002a,b,c, Wang 1996, Johnstone and Silverman 1997, Yang 2001, Li and Xiao 2007, Kulik and Raimondo 2009, Beran and Shumeyko 2011a; also see Ghosh and Draghicescu 2002a,b for quantile smoothing). For kernel and local polynomial trend estimation, data driven iterative algorithms are available (Ray and Tsay 1997, Beran and Feng 2002a,b, Ghosh and Draghicescu 2002b). For wavelet thresholding, most results deal with the minimax approach. Often, minimax solutions are not suitable for a concrete data analysis. Beran and Shumeyko (2011a) therefore suggest a data driven wavelet method and derive asymptotically optimal tuning parameters (also see Li and Xiao 2007 for related results). This is discussed in the next section (section 2). The derivation of the asymptotic mean squared error leads to a natural decomposition of the trend estimator into a smooth and a discontinuous component. An application of this decomposition is testing for discontinuities in the trend function (Beran and Shumeyko 2011b). This is discussed in section 3 below. To obtain a method that is applicable to relatively short time series, a bootstrap statistic and an algorithm for computing its finite sample distribution are defined. The blockwise approach used here is similar to Lahiri (1993) (also compare with Percival et al. 2000). The validity and consistency of the test procedure is shown under the assumption of Gaussian residuals. A generalization to subordinated processes is possible but not pursued here in detail. For literature on structural breaks in the long-memory context see for example references in Sibbertsen (2004) and Banerjee and Urgab (2005).