123 research outputs found
Continuous invertibility and stable QML estimation of the EGARCH(1,1) model
We introduce the notion of continuous invertibility on a compact set for
volatility models driven by a Stochastic Recurrence Equation (SRE). We prove
the strong consistency of the Quasi Maximum Likelihood Estimator (QMLE) when
the optimization procedure is done on a continuously invertible domain. This
approach gives for the first time the strong consistency of the QMLE used by
Nelson in \cite{nelson:1991} for the EGARCH(1,1) model under explicit but non
observable conditions. In practice, we propose to stabilize the QMLE by
constraining the optimization procedure to an empirical continuously invertible
domain. The new method, called Stable QMLE (SQMLE), is strongly consistent when
the observations follow an invertible EGARCH(1,1) model. We also give the
asymptotic normality of the SQMLE under additional minimal assumptions
Deviation inequalities for sums of weakly dependent time series
In this paper we give new deviation inequalities of Bernstein's type for the
partial sums of weakly dependent time series. The loss from the independent
case is studied carefully. We give non mixing examples such that dynamical
systems and Bernoulli shifts for whom our deviation inequalities hold. The
proofs are based on the blocks technique and different coupling arguments.Comment: 14 page
Exponential inequalities for unbounded functions of geometrically ergodic Markov chains. Applications to quantitative error bounds for regenerative Metropolis algorithms
The aim of this note is to investigate the concentration properties of
unbounded functions of geometrically ergodic Markov chains. We derive
concentration properties of centered functions with respect to the square of
the Lyapunov's function in the drift condition satisfied by the Markov chain.
We apply the new exponential inequalities to derive confidence intervals for
MCMC algorithms. Quantitative error bounds are providing for the regenerative
Metropolis algorithm of [5]
Model selection for weakly dependent time series forecasting
Observing a stationary time series, we propose a two-step procedure for the
prediction of the next value of the time series. The first step follows machine
learning theory paradigm and consists in determining a set of possible
predictors as randomized estimators in (possibly numerous) different predictive
models. The second step follows the model selection paradigm and consists in
choosing one predictor with good properties among all the predictors of the
first steps. We study our procedure for two different types of bservations:
causal Bernoulli shifts and bounded weakly dependent processes. In both cases,
we give oracle inequalities: the risk of the chosen predictor is close to the
best prediction risk in all predictive models that we consider. We apply our
procedure for predictive models such as linear predictors, neural networks
predictors and non-parametric autoregressive
The cluster index of regularly varying sequences with applications to limit theory for functions of multivariate Markov chains
We introduce the cluster index of a multivariate regularly varying stationary
sequence and characterize the index in terms of the spectral tail process. This
index plays a major role in limit theory for partial sums of regularly varying
sequences. We illustrate the use of the cluster index by characterizing
infinite variance stable limit distributions and precise large deviation
results for sums of multivariate functions acting on a stationary Markov chain
under a drift condition
Convergence rates for density estimators of weakly dependent time series
Assuming that is a vector valued time series with a common
marginal distribution admitting a density , our aim is to provide a wide
range of consistent estimators of . We consider different methods of
estimation of the density as kernel, projection or wavelets ones. Various cases
of weakly dependent series are investigated including the Doukhan & Louhichi
(1999)'s -weak dependence condition, and the -dependence of
Dedecker & Prieur (2005). We thus obtain results for Markov chains, dynamical
systems, bilinear models, non causal Moving Average... From a moment inequality
of Doukhan & Louhichi (1999), we provide convergence rates of the term of error
for the estimation with the \L^q loss or almost surely, uniformly on compact
subsets
Precise large deviations for dependent regularly varying sequences
We study a precise large deviation principle for a stationary regularly
varying sequence of random variables. This principle extends the classical
results of A.V. Nagaev (1969) and S.V. Nagaev (1979) for iid regularly varying
sequences. The proof uses an idea of Jakubowski (1993,1997) in the context of
centra limit theorems with infinite variance stable limits. We illustrate the
principle for \sv\ models, functions of a Markov chain satisfying a polynomial
drift condition and solutions of linear and non-linear stochastic recurrence
equations
Fast rates in learning with dependent observations
In this paper we tackle the problem of fast rates in time series forecasting
from a statistical learning perspective. In a serie of papers (e.g. Meir 2000,
Modha and Masry 1998, Alquier and Wintenberger 2012) it is shown that the main
tools used in learning theory with iid observations can be extended to the
prediction of time series. The main message of these papers is that, given a
family of predictors, we are able to build a new predictor that predicts the
series as well as the best predictor in the family, up to a remainder of order
. It is known that this rate cannot be improved in general. In this
paper, we show that in the particular case of the least square loss, and under
a strong assumption on the time series (phi-mixing) the remainder is actually
of order . Thus, the optimal rate for iid variables, see e.g. Tsybakov
2003, and individual sequences, see \cite{lugosi} is, for the first time,
achieved for uniformly mixing processes. We also show that our method is
optimal for aggregating sparse linear combinations of predictors
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