Two-step wavelet-based estimation for Gaussian mixed fractional processes

A Gaussian mixed fractional process $$\{Y(t)\}_{t \in {\mathbb {R}}} = \{PX(t)\}_{t \in {\mathbb {R}}}$$ { Y ( t ) } t ∈ R = { P X ( t ) } t ∈ R is a multivariate stochastic process obtained by pre-multiplying a vector of independent, Gaussian fractional process entries X by a nonsingular matrix P. It is interpreted that Y is observable, while X is a hidden process occurring in an (unknown) system of coordinates P. Mixed processes naturally arise as approximations to solutions of physically relevant classes of multivariate fractional stochastic differential equations under aggregation. We propose a semiparametric two-step wavelet-based method for estimating both the demixing matrix $$P^{-1}$$ P - 1 and the memory parameters of X. The asymptotic normality of the estimator is established both in continuous and discrete time. Monte Carlo experiments show that the estimator is accurate over finite samples, while being very computationally efficient. As an application, we model a bivariate time series of annual tree ring width measurements.