A statistical test for the time constancy of scaling exponents

A statistical test is described for determining if scaling exponents vary over time. It is applicable to diverse scaling phenomena including long range dependence and exactly self-similar processes in a uniform framework, without the need for prior knowledge of the type in question. It is based on the special properties of wavelet-based estimates of the scaling exponent, strongly motivating an idealised inference problem: the equality or otherwise of means of independent Gaussian variables with known variances. A uniformly most powerful invariant test exists for this problem and is described. A separate UMPI test is also given for when the scaling exponent undergoes a level change. The power functions of both tests are given explicitly and compared. Using simulation the effect in practice of deviations from the idealisations made of the statistical properties of the wavelet detail coefficients are analysed and found to be small. The tests inherit the significant robustness and computational advantages of the underlying wavelet-based estimator. A detailed methodology is given describing its use in practical situations. The use and benefits of the test are illustrated on the Bellcore Ethernet data sets. Keywords— Self-similarity, long-range dependence, scaling exponent, wavelets, hypothesis testing, stationarity, telecommunications networks.