Scale parameter estimation of discrete scale invariant processes

Estimating the scale parameter of a continuous-time discrete scale invariant (DSI) process is one of the fundamental problems in the literature. We present an efficient estimation method which is based on transforming the DSI process into a vector-valued self-similar process and allows us to obtain the structure of the covariance matrix, which is the product of a scale and a block-Toeplitz matrices, and its block size depends on the unknown scale parameter. Therefore, we sweep the block size and obtain the maximum likelihood (ML) estimate of the scale parameter. We show that, the ML estimator does not directly solve the scale estimation problem. Hence, we penalize the likelihood following an information theoretic approach. The performance of the estimation method is studied via simulation. Finally this method is applied to the real data of S$\&$P500 and Dow Jones indices for some special periods.