2,753 research outputs found
Efficient likelihood estimation in state space models
Motivated by studying asymptotic properties of the maximum likelihood
estimator (MLE) in stochastic volatility (SV) models, in this paper we
investigate likelihood estimation in state space models. We first prove, under
some regularity conditions, there is a consistent sequence of roots of the
likelihood equation that is asymptotically normal with the inverse of the
Fisher information as its variance. With an extra assumption that the
likelihood equation has a unique root for each , then there is a consistent
sequence of estimators of the unknown parameters. If, in addition, the supremum
of the log likelihood function is integrable, the MLE exists and is strongly
consistent. Edgeworth expansion of the approximate solution of likelihood
equation is also established. Several examples, including Markov switching
models, ARMA models, (G)ARCH models and stochastic volatility (SV) models, are
given for illustration.Comment: With the comments by Jens Ledet Jensen and reply to the comments.
Published at http://dx.doi.org/10.1214/009053606000000614;
http://dx.doi.org/10.1214/09-AOS748A; http://dx.doi.org/10.1214/09-AOS748B in
the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Regenerative block empirical likelihood for Markov chains
Empirical likelihood is a powerful semi-parametric method increasingly
investigated in the literature. However, most authors essentially focus on an
i.i.d. setting. In the case of dependent data, the classical empirical
likelihood method cannot be directly applied on the data but rather on blocks
of consecutive data catching the dependence structure. Generalization of
empirical likelihood based on the construction of blocks of increasing
nonrandom length have been proposed for time series satisfying mixing
conditions. Following some recent developments in the bootstrap literature, we
propose a generalization for a large class of Markov chains, based on small
blocks of various lengths. Our approach makes use of the regenerative structure
of Markov chains, which allows us to construct blocks which are almost
independent (independent in the atomic case). We obtain the asymptotic validity
of the method for positive recurrent Markov chains and present some simulation
results
Adventures in Invariant Theory
We provide an introduction to enumerating and constructing invariants of
group representations via character methods. The problem is contextualised via
two case studies arising from our recent work: entanglement measures, for
characterising the structure of state spaces for composite quantum systems; and
Markov invariants, a robust alternative to parameter-estimation intensive
methods of statistical inference in molecular phylogenetics.Comment: 12 pp, includes supplementary discussion of example
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