On the invertibility of time series models

A generalized definition of invertibility is proposed and applied to linear, non-linear and bilinear models. It is shown that some recently studied non-linear models are not invertible, but conditions for invertibility can be achieved for the other models.     

On the estimation of nonlinear time series models

In the present paper, a framework for parametric estimation in nonlinear time series is developed. Strong consistency and asymptotic normality of minimum Hellinger distance estimates for a determined class of nonlinear models are investigated. The main Interest for these estimates is motivated by their robustness under perturbations as it has been emphazized in Beran [2]. The first part of the paper is devoted to the study of some probabilistic properties which ensure the existence and the optimal properties of the estimates     

On categorical time series models with covariates

We study the problem of stationarity and ergodicity for autoregressive multinomial logistic time series models which possibly include a latent process and are defined by a GARCH-type recursive equation. We improve considerably upon the existing conditions about stationarity and ergodicity of those models. Proofs are based on theory developed for chains with complete connections. A useful coupling technique is employed for studying ergodicity of infinite order finite-state stochastic processes which generalize finite-state Markov chains. Furthermore, for the case of finite order Markov chains, we discuss ergodicity properties of a model which includes strongly exogenous but not necessarily bounded covariates.     

On pricing derivatives under nonlinear time series models

In this note, we summarize some of our recent works on pricing derivative securities under nonlinear time series models. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new approach to the problem of estimating spectral parameters of non-stationary time series models

A modified weighted least square approach for estimating the time-dependent parameters of non-stationary autoregressive models is proposed. Moreover a modified frequency domain approach for estimating the parameters of non-stationary time-series models is also presented. The original and modified weighted least square procedures have been compared numerically.     

On some sampling schemes for estimating the parameters of a continuous time series

Summary In estimating the mean of certain stationary processes it is shown that it is better to sample at fixed equi-spaced time intervals than to sample randomly according to a renewal process. On the other hand it is shown that the estimation of autocorrelation is sometimes better accomplished by random sampling.     

Exact filtering in conditionally Markov switching hidden linear models

In the classical setup of Markov switching hidden linear systems, the exact evaluation of optimal filters requires computations with complexity increasing exponentially with respect to the number of observations. In the present article, we propose a new family of models which overcome this difficulty, and render practically feasible these calculations.     

High-level dependence in time series models

We present several notions of high-level dependence for stochastic processes, which have appeared in the literature. We calculate such measures for discrete and continuous-time models, where we concentrate on time series with heavy-tailed marginals, where extremes are likely to occur in clusters. Such models include linear models and solutions to random recurrence equations; in particular, discrete and continuous-time moving average and (G)ARCH processes. To illustrate our results we present a small simulation study.     

Estimation in nonlinear time series models

A general framework for analyzing estimates in nonlinear time series is developed. General conditions for strong consistency and asymptotic normality are derived both for conditional least squares and maximum likelihood types estimates. Ergodie strictly stationary processes are studied in the first part and certain nonstationary processes in the last part of the paper. Examples are taken from most of the usual classes of nonlinear time series models.     

On quasi-static models hidden in Maxwell's equations

In this paper we introduce the electromagnetic quasi-static models in a simple but meaningful way, relying on the dimensional analysis of Maxwell's equations. This analysis puts in evidence the three characteristic times of an electromagnetic phenomenon. It allows to define the range of validity of well-known models, such as the eddy-current (MQS) or the electroquasistatic (EQS) ones, and thus their pertinence to describe a given phenomenon. The role of the so-called “small parameters” of a model is explained in detail for two classical examples, namely a capacitor and a solenoid. We show how the MQS and EQS models result from having replaced fields by their first order truncations of Taylor expansions with respect to these small parameters. We finally investigate the connection between quasi-static models and circuit theory, clarifying the role of the fields with respect to classical circuit elements, and provide an example of application to study the electromagnetic fields in a simple case.     

Recovery of periodicities hidden in heavy‐tailed noise

We address a parametric joint detection‐estimation problem for discrete signals of the form , , with an additive noise represented by independent centered complex random variables . The distributions of are assumed to be unknown, but satisfying various sets of conditions. We prove that in the case of a heavy‐tailed noise it is possible to construct asymptotically strongly consistent estimators for the unknown parameters of the signal, i.e., frequencies , their number N, and complex coefficients . For example, one of considered classes of noise is the following: are independent identically distributed random variables with and . The construction of estimators is based on detection of singularities of anti‐derivatives for Z‐transforms and on a two‐level selection procedure for special discretized versions of superlevel sets. The consistency proof relies on the convergence theory for random Fourier series.     

Estimators of covariances in time series models

The theory of Minimum Norm Quadratic Estimators for estimating variances and covariances is applied to show that some commonly used estimators of covariances in time series models are easily derived using the above principle. In applying the theory MINQE, it is observed that no unbiased estimator exists in the class of invariant quadratics.     

Maximum likelihood estimator for hidden Markov models in continuous time

The paper studies large sample asymptotic properties of the Maximum Likelihood Estimator (MLE) for the parameter of a continuous time Markov chain, observed in white noise. Using the method of weak convergence of likelihoods due to Ibragimov and Khasminskii (Statistical estimation, vol 16 of Applications of mathematics. Springer-Verlag, New York), consistency, asymptotic normality and convergence of moments are established for MLE under certain strong ergodicity assumptions on the chain. This article has been written during the author’s visit at Laboratoire de Statistique et Processus, Universite du Maine, France, supported by the Chateaubriand fellowship.     

On series in linear topological spaces

The main result is that in every complete locally-bounded linear topological space there exist series which are unconditionally yet not absolutely convergent. Relations between absolute, unconditional and metric convergence of series are studied. The research reported in this document has been sponsored in part by the Air Force Office of Scientific Research, OAR through the European Office, Aerospace Research, United States Air Force.     

Linear relations in time series models. I

A multiple time series is defined as the sum of an autoregressive process on a line and independent Gaussian white noise on a hyperplane that goes through the origin and intersects the line at a single point. This process is a multiple autoregressive time series in which the regression matrices satisfy suitable conditions. It is shown that the maximum likelihood estimates of the line and the autoregression coefficients can be obtained as the values that minimize a given function, and that the remaining maximum likelihood estimates can be computed as simple functions of the first ones. It is also shown that the maximum likelihood estimates are equivariant with respect to the group of bijective linear transformations.     

Linear relations in time series models. II

In this paper an asymptotic theory is developed for a new time series model which was introduced in a previous paper [5]. An algorithm for computing estimates of the parameters of this time series model is given, and it is shown that these estimators are asymptotically efficient in the sense that they have the same asymptotic distribution as the maximum likelihood estimators.     

On the calculation of cumulants of estimators arising from a linear time series regression model

Bounds for higher-order cumulants of statistics arising from a linear time series regression model are investigated. A result given in Brillinger is proved and extended. The bounds permit derivation of asymptotic moments and asymptotic normality for estimators of parameters in the model. Two examples are given as illustrations.