Abstract
In this work, we propose a Bayesian methodology to make inferences for the memory parameter and other characteristics under non-standard assumptions for a class of stochastic processes. This class generalizes the Gamma-modulated process, with trajectories that exhibit long memory behavior, as well as decreasing variability as time increases. Different values of the memory parameter influence the speed of this decrease, making this heteroscedastic model very flexible. Its properties are used to implement an approximate Bayesian computation and MCMC scheme to obtain posterior estimates. We test and validate our method through simulations and real data from the big earthquake that occurred in 2010 in Chile.
Highlights
Diffusion processes have been a cornerstone of stochastic modeling of time series data, in areas such as finance [1] and hydrology [2]
Many extensions to the classic diffusion model have been developed in recent years, addressing such diverse issues as asymmetry, kurtosis, heteroscedasticity and long memory; see, for instance, [3]
The increments of a diffusion model are taken as independent Gaussian random variables, making the process a Brownian motion
Summary
Diffusion processes have been a cornerstone of stochastic modeling of time series data, in areas such as finance [1] and hydrology [2]. The memory parameter was assumed to be known and fixed, with some particular cases, such as the standard Brownian motion and the Student process The latter one is a generalization of the Student process previously presented in [5], the marginals of which have a t-Student distribution with fixed degrees of freedom and a long memory structure. We enlarge the parameter space considering that the memory parameter is unknown, provided we have a prior distribution on it This extension allows flexibility for the dependence structure of the process, where the Brownian motion and the G-M process become particular cases. We will focus on estimation procedures for long-range memory stochastic models from a Bayesian perspective Other parameters, such as location and dispersion, are considered.
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