Generalized Likelihood Ratio Test for Detection of Multivariate DSI Processes

This paper provides a new method in detecting multivariate discrete scale invariant (DSI) processes using an asymptotic generalized likelihood ratio test (GLRT). We consider two hypothesis tests: 1) Is a multivariate process, DSI or is it self-similar? 2) Is a multivariate process, DSI or is it nonstationary?. Then, using the asymptotic GLRT, the DSI behaviour can be detected. In this method, by imposing some flexible sampling scheme, we provide some discretization of continuous time discrete scale invariant (DSI) processes. Then, the relationship between a discrete-time DSI process and a corresponding multidimensional self-similar process, enables us to formulate the problem as a test for covariance structure of the processes. For DSI and self-similar processes, the covariance matrices are as a product of scale matrices to a block-Toeplitz matrix, in which there is no a closed form of maximum likelihood for such matrices. So, by considering the asymptotic case, where the block-Toeplitz matrix converges to a block-circulant matrix, the asymptotic GLRT is derived. To clarify the proposed method, an example as a multivariate simple Brownian motion is presented and its simulations are provided. Also the performance of the method is studied on the S$\&$P500 and Daw Jones indices for some special periods.