1A Statistical Test for the Time Constancy of Scaling Exponents

Abstract—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 ex-actly 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 prob-lem 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 com-pared. 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 com-putational advantages of the underlying wavelet-based esti-mator. 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