### Testing for homogeneity of variance in the wavelet domain [chapter]

Olaf Kouamo, Eric Moulines, Francois Roueff
2010 Lecture Notes in Statistics
The danger of confusing long-range dependence with non-stationarity has been pointed out by many authors. Finding an answer to this difficult question is of importance to model time-series showing trend-like behavior, such as river runoff in hydrology, historical temperatures in the study of climates changes, or packet counts in network traffic engineering. The main goal of this paper is to develop a test procedure to detect the presence of non-stationarity for a class of processes whose K-th
more » ... der difference is stationary. Contrary to most of the proposed methods, the test procedure has the same distribution for short-range and long-range dependence covariance stationary processes, which means that this test is able to detect the presence of non-stationarity for processes showing long-range dependence or which are unit root. The proposed test is formulated in the wavelet domain, where a change in the generalized spectral density results in a change in the variance of wavelet coefficients at one or several scales. Such tests have been already proposed in Whitcher et al. (2001), but these authors do not have taken into account the dependence of the wavelet coefficients within scales and between scales. Therefore, the asymptotic distribution of the test they have proposed was erroneous; as a consequence, the level of the test under the null hypothesis of stationarity was wrong. In this contribution, we introduce two test procedures, both using an estimator of the variance of the scalogram at one or several scales. The asymptotic distribution of the test under the null is rigorously justified. The pointwise consistency of the test in the presence of a single jump in the general spectral density is also be presented. A limited Monte-Carlo experiment is performed to illustrate our findings. H 0 , when the asymptotic level is set to 0.05. White noise n 512 1024 2048 4096 8192 J = 3 KSM 0.02 0.01 0.03 0.02 0.02 J = 3 CV M 0.05 0.045 0.033 0.02 0.02 J = 4 KSM 0.047 0.04 0.04 0.02 0.02 J = 4 CV M 0.041 0.02 0.016 0.016 0.01 J = 5 KSM 0.09 0.031 0.02 0.025 0.02 J = 5 CV M 0.086 0.024 0.012 0.012 0.02 Table 1.3. Empirical level of KSM − CVM for a white noise. n 512 1024 2048 4096 8192 J = 3 KSM 0.028 0.012 0.012 0.012 0.02 J = 3 CV M 0.029 0.02 0.016 0.016 0.01 J = 4 KSM 0.055 0.032 0.05 0.025 0.02 J = 4 CV M 0.05 0.05 0.03 0.02 0.02 J = 5 KSM 0.17 0.068 0.02 0.02 0.02 J = 5 CV M 0.13 0.052 0.026 0.021 0.02 Table 1.4. Empirical level of KSM − CVM for a M A(q) process. AR(1)[φ = 0.9] n 512 1024 2048 4096 8192 J = 3 KSM 0.083 0.073 0.072 0.051 0.04 J = 3 CV M 0.05 0.05 0.043 0.032 0.03 J = 4 KSM 0.26 0.134 0.1 0.082 0.073 J = 4 CV M 0.14 0.092 0.062 0.04 0.038 J = 5 KSM 0.547 0.314 0.254 0.22 0.11 J = 5 CV M 0.378 0.221 0.162 0.14 0.093