Estimation for L\'evydriven CARMA Processes

Yu "Vicky" Yang
Ph.D. Candidate, Department of Statistics, Colorado State University
Friday, July 13, 2007
2:00 p.m.
006 Statistics
ABSTRACT
This thesis explores parameter estimation for
Levydriven continuoustime autoregressive moving average
(CARMA) processes, using uniformly and closely spaced
discretetime observations.
Specifically, we focus on developing estimation techniques and
asymptotic properties of the estimators for three particular
families of Levydriven 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 closelyspaced discretetime observations. We apply the
results to both linear and nonlinear autoregressive processes.
For the second family, nonnegative Levydriven
OrnesteinUhlenbeck processes, we take advantage of the
nonnegativity 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, nonnegative Levydriven 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.
