Estimation for L\'evy-driven CARMA Processes
Yu "Vicky" Yang

Ph.D. Candidate, Department of Statistics, Colorado State University

Friday, July 13, 2007
2:00 p.m.
006 Statistics


This thesis explores parameter estimation for Levy-driven continuous-time autoregressive moving average (CARMA) processes, using uniformly and closely spaced discrete-time observations.

Specifically, we focus on developing estimation techniques and asymptotic properties of the estimators for three particular
families of Levy-driven CARMA processes. Estimation for the
first family, Gaussian autoregressive processes, was developed by
deriving exact conditional maximum likelihood estimators of the
parameters under the assumption that the process is observed
continuously. The resulting estimates are expressed in terms of
stochastic integrals which are then approximated using the
available closely-spaced discrete-time observations. We apply the
results to both linear and non-linear autoregressive processes.
For the second family, non-negative Levy-driven
Ornestein-Uhlenbeck processes, we take advantage of the
non-negativity of the increments of the driving Levy processes
to derive a highly efficient estimation procedure for the
autoregressive coefficient when observations are available at
uniformly spaced times. Asymptotic properties of the estimator
are also studied and a procedure for obtaining estimates of the
increments of the driving Levy process is developed. These
estimated increments are important for identifying the nature of
the driving Levy process and for estimating its parameters. For
the third family, non-negative Levy-driven CARMA processes, we
estimate the coefficients by maximizing the Gaussian likelihood
of the observations and discuss the asymptotic properties of the
estimators. We again show how to estimate the increments of the
background driving Levy process and hence to estimate the
parameters of the Levy process itself. We assess the
performance of our estimation procedures by simulations and use
them to fit models to real data sets in order to determine how
the theory applies in practice.



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