49,153 research outputs found
Invariance principle for stochastic processes with short memory
In this paper we give simple sufficient conditions for linear type processes
with short memory that imply the invariance principle. Various examples
including projective criterion are considered as applications. In particular,
we treat the weak invariance principle for partial sums of linear processes
with short memory. We prove that whenever the partial sums of innovations
satisfy the --invariance principle, then so does the partial sums of its
corresponding linear process.Comment: Published at http://dx.doi.org/10.1214/074921706000000734 in the IMS
Lecture Notes Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
Invariance principles for linear processes with application to isotonic regression
In this paper, we prove maximal inequalities and study the functional central
limit theorem for the partial sums of linear processes generated by dependent
innovations. Due to the general weights, these processes can exhibit long-range
dependence and the limiting distribution is a fractional Brownian motion. The
proofs are based on new approximations by a linear process with martingale
difference innovations. The results are then applied to study an estimator of
the isotonic regression when the error process is a (possibly long-range
dependent) time series.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ273 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Distributed Constrained Recursive Nonlinear Least-Squares Estimation: Algorithms and Asymptotics
This paper focuses on the problem of recursive nonlinear least squares
parameter estimation in multi-agent networks, in which the individual agents
observe sequentially over time an independent and identically distributed
(i.i.d.) time-series consisting of a nonlinear function of the true but unknown
parameter corrupted by noise. A distributed recursive estimator of the
\emph{consensus} + \emph{innovations} type, namely , is
proposed, in which the agents update their parameter estimates at each
observation sampling epoch in a collaborative way by simultaneously processing
the latest locally sensed information~(\emph{innovations}) and the parameter
estimates from other agents~(\emph{consensus}) in the local neighborhood
conforming to a pre-specified inter-agent communication topology. Under rather
weak conditions on the connectivity of the inter-agent communication and a
\emph{global observability} criterion, it is shown that at every network agent,
the proposed algorithm leads to consistent parameter estimates. Furthermore,
under standard smoothness assumptions on the local observation functions, the
distributed estimator is shown to yield order-optimal convergence rates, i.e.,
as far as the order of pathwise convergence is concerned, the local parameter
estimates at each agent are as good as the optimal centralized nonlinear least
squares estimator which would require access to all the observations across all
the agents at all times. In order to benchmark the performance of the proposed
distributed estimator with that of the centralized nonlinear
least squares estimator, the asymptotic normality of the estimate sequence is
established and the asymptotic covariance of the distributed estimator is
evaluated. Finally, simulation results are presented which illustrate and
verify the analytical findings.Comment: 28 pages. Initial Submission: Feb. 2016, Revised: July 2016,
Accepted: September 2016, To appear in IEEE Transactions on Signal and
Information Processing over Networks: Special Issue on Inference and Learning
over Network
Quasi maximum likelihood estimation for strongly mixing state space models and multivariate L\'evy-driven CARMA processes
We consider quasi maximum likelihood (QML) estimation for general
non-Gaussian discrete-ime linear state space models and equidistantly observed
multivariate L\'evy-driven continuoustime autoregressive moving average
(MCARMA) processes. In the discrete-time setting, we prove strong consistency
and asymptotic normality of the QML estimator under standard moment assumptions
and a strong-mixing condition on the output process of the state space model.
In the second part of the paper, we investigate probabilistic and analytical
properties of equidistantly sampled continuous-time state space models and
apply our results from the discrete-time setting to derive the asymptotic
properties of the QML estimator of discretely recorded MCARMA processes. Under
natural identifiability conditions, the estimators are again consistent and
asymptotically normally distributed for any sampling frequency. We also
demonstrate the practical applicability of our method through a simulation
study and a data example from econometrics
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