New methods of estimation of long-memory models with an application in climatology
Abstract
The estimation of long-memory processes has been studied from different perspectives: non-parametric, parametric and bayesian. From a general point of view, we consider a stationary gaussian process with explicit parametric spectral density. Under some conditions on its autocovariance function, we defined a GMM estimator that satisfies consistency and asymptotic normality, using previous results on ergodicity and the Breuer-Major theorem. This result is applied to the joint estimation of the drift and Hurst (H) parameters of a stationary Ornstein-Uhlenbeck (fOU) process driven by a fractional Brownian motion. The asymptotic normality of its GMM estimator applies for any H in (0,3/4), but using Malliavin calculus techniques we extend it to the case H=3/4. This parametric technique is also illustrated with the long-memory estimation of the fractional Gaussian noise. From a bayesian perspective, the estimation of long-memory parameters is more related to filtering problems. As an application of this particularity, we performed a reconstruction of the Northern Hemisphere (NH) temperature anomalies over the past millennium by combining both the temperature proxies and external forcings while taking the possible long memory features of the involved stochastic processes into account, using a Hierarchical bayesian model. Our reconstruction is based on two linear equations, one in which the unobserved temperature series is linearly related to the three main external forcings, and another in which the observed temperature proxy data (tree rings, ice cores, and others) which is linearly related to the unobserved temperatures. Uncertainty is modeled through a long-memory fractional Gaussian noise. Our MCMC results give us reconstructions of the past unobserved land-air temperature anomalies and combined land-and-marine anomalies over the period 1000-1899, together with a precise description of the reconstruction uncertainty in the form of an empirical posterior distribution for the temperature series.
Degree
Ph.D.
Advisors
Li, Purdue University.
Subject Area
Applied Mathematics|Statistics|Paleoclimate Science
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