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

TL;DR

This paper introduces a semiparametric wavelet-based method for estimating the structure and memory parameters of mixed Gaussian fractional processes, with proven asymptotic properties and practical applications.

Contribution

It develops a novel two-step wavelet-based estimation technique for mixed Gaussian fractional processes, including asymptotic analysis and real-world applications.

Findings

Estimators are asymptotically normal in continuous and discrete time.

Finite sample performance is comparable to parametric methods.

Method is computationally efficient and applicable to real data.

Abstract

A mixed Gaussian fractional process $\{Y(t)\}_{t \in {\Bbb R}} = \{PX(t)\}_{t \in {\Bbb R}}$ 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 SDEs under aggregation. We propose a semiparametric two-step wavelet-based method for estimating both the demixing matrix $P^{-1}$ and the memory parameters of $X$. The asymptotic normality of the estimators is established both in continuous and discrete time. Monte Carlo experiments show that the finite sample estimation performance is comparable to that of parametric methods, while being very computationally efficient. As applications, we model a bivariate time series of annual tree ring width measurements, and establish the asymptotic normality of the eigenstructure of sample wavelet matrices.