Applied Maths Seminar: Zou -- Some statistical inference methods for dynamical systems

Nan Zou (Macquarie University) will be giving a seminar on Wednesday 7 October at 12pm
in Carslaw 451.  We will go to lunch afterwards with the speaker (all welcome, free for
students).  

Title: Some statistical inference methods for dynamical systems 

Abstract: In many applications, the transformation function governing a dynamical system
is unknown, and the available observations consist of a single, finite-length orbit.
This talk considers statistical inference, e.g., confidence intervals and hypothesis
tests, of unknown quantities associated with the dynamical system using these
observations.  It will first focus on the statistical inference on spatial averages ---
the integrals of observables with respect to an ergodic invariant measure --- using time
averages as estimators.  A key challenge is that when the central limit theorem/Gaussian
approximation applies, the scale of the Gaussian distribution is controlled by a
diffusion parameter generally unknown in practice.  

To address this challenge, this talk will discuss three methods from the time-series
analysis literature and will examine their behaviours in the dynamical system setting.
The first directly estimates the diffusion parameter from the observations and then
substitutes the estimate into the Gaussian distribution.  This talk will review the
estimator and discuss, under the spectral gap assumption, the consistency and a
conjectured central limit theorem for this estimator based on a large-block-small-block
technique.  The second method, self-normalisation, rescales the time average without
explicitly estimating the diffusion parameter; this talk will introduce the method and
illustrate its simulation performance.  The third method, bootstrap, seeks to
approximate both the scale and the shape of the sampling distribution of the time
average.  This talk will introduce a dynamical-system bootstrap and discuss its
higher-order accuracy over the Gaussian approximation.  If time permits, this talk will
also touch on possible extensions from spatial averages to more general functionals of
the invariant measure.  This involves joint work with B.  Bajar, K.  Fernando and G.
Sofronov.