Periodically correlated autoregressive Hilbertian processes

We consider periodically correlated autoregressive processes in Hilbert spaces. Our studies on these processes involve existence, covariance structure, estimation of the covariance operators, strong law of large numbers and central limit theorem.     

Ergodicity of nonlinear first order autoregressive models

Criteria are derived for ergodicity and geometric ergodicity of Markov processes satisfyingX _n+1 =f(X _n )+(X _n ) _n+1 , wheref, are measurable, { _n } are i.i.d. with a (common) positive density,E| _n |>. In the special casef(x)/x has limits, , asx– andx+, respectively, it is shown that <1, <1, <1 is sufficient for geometric ergodicity, and that <-1, 1, 1 is necessary for recurrence.     

Some laws of the iterated logarithm in Hilbertian autoregressive models

We consider the law of the iterated logarithm for the empirical covariance of Hilbertian autoregressive processes. As an application, we obtain laws of the iterated logarithm for the eigenvalues and associated projectors of the empirical covariance.     

Spatial autoregressive and moving average Hilbertian processes

This paper addresses the introduction and study of structural properties of Hilbert-valued spatial autoregressive processes (SARH(1) processes), and Hilbert-valued spatial moving average processes (SMAH(1) processes), with innovations given by two-parameter (spatial) matingale differences. For inference purposes, the conditions under which the tensorial product of standard autoregressive Hilbertian (ARH(1)) processes (respectively, of standard moving average Hilbertian (MAH(1)) processes) is a standard SARH(1) process (respectively, it is a standard SMAH(1) process) are studied. Examples related to the spatial functional observation of two-parameter Markov and diffusion processes are provided. Some open research lines are described in relation to the formulation of SARMAH processes, as well as General Spatial Linear Processes in Functional Spaces.     

Estimation in threshold autoregressive models with correlated innovations

Large sample statistical analysis of threshold autoregressive models is usually based on the assumption that the underlying driving noise is uncorrelated. In this paper, we consider a model, driven by Gaussian noise with geometric correlation tail and derive a complete characterization of the asymptotic distribution for the Bayes estimator of the threshold parameter.     

On unit roots for spatial autoregressive models

In this paper we consider the unit root problem for one rather simple autoregressive model Y_t,s=aY_t-1,s+bY_t,s-1+?_t,s on a two-dimensional lattice. We show that the growth of variance of Y_t,s is essentially different from corresponding growth in the unit root case for AR(1) or AR(2) time series models. We also show that the dimension of the lattice plays an important role: the growth of variance of autoregressive field on a d-dimensional lattice is different for d=2,3 and d≥4.     

On the marginal distribution of a first order autoregressive process

The question of what marginal distributions are possible for a first order autoregressive process is addressed. Results concerning the possible multimodality of the marginal distribution are obtained.     

Finite-sample properties of estimators for first and second order autoregressive processes

Statistical Inference for Stochastic Processes - The class of autoregressive (AR) processes is extensively used to model temporal dependence in observed time series. Such models are easily...     

Asymptotic properties of the maximum likelihood estimate in the first order autoregressive process

Summary In this paper we obtain an asymptotic expansion of the distribution of the maximum likelihood estimate (MLE) based onT observations from the first order Gaussian process up to the term of orderT ⁻¹. The expansion is used to compare with a generalized estimate including the least square estimate (LSE) , based on the asymptotic probabilities around the true value of the estimates up to the terms of orderT ⁻¹. It is shown that (or the modified MLE ) is better than (or the modified estimate ). Further, we note that does not attain the bound for third order asymptotic median unbiased estimates.     

Bernstein–Frechet inequalities for the parameter of the first order autoregressive process

The autoregressive process takes an important part in predicting problems leading to decision making. In practice, we use the least squares method to estimate the parameter of the autoregressive process. In the case of the first order autoregressive process, we know that the least squares estimator converges in probability to the unknown parameter θ. In this Note, we show that the least squares estimator converges almost completely to θ and so we construct the inequalities of type Bernstein–Frechet for the coefficient of the first order autoregressive process. Using these inequalities a confidence interval is then obtained. To cite this article: A. Dahmani, M. Tari, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Estimation of spatial autoregressive models with measurement error for large data sets

Maximum likelihood (ML) estimation of spatial autoregressive models for large spatial data sets is well established by making use of the commonly sparse nature of the contiguity matrix on which spatial dependence is built. Adding a measurement error that naturally separates the spatial process from the measurement error process are not well established in the literature, however, and ML estimation of such models to large data sets is challenging. Recently a reduced rank approach was suggested which re-expresses and approximates such a model as a spatial random effects model (SRE) in order to achieve fast fitting of large data sets by fitting the corresponding SRE. In this paper we propose a fast and exact method to accomplish ML estimation and restricted ML estimation of complexity of \(O(n^{3/2})\) operations when the contiguity matrix is based on a local neighbourhood. The methods are illustrated using the well known data set on house prices in Lucas County in Ohio.     

First order autoregressive markov processes

This paper is concerned with establishing conditions under which finite (and then countably infinite) stationary Markov chains have first order autoregressive representations.     

On the bias of the least squares estimator for the first order autoregressive process

The paper provides an exact formula for the bias of the parameter estimator of the first order autoregressive process and derives the asymptotic bias.     

Truncated sequential estimation of the parameter of a first order autoregressive process with dependent noises

For a first-order non-explosive autoregressive process with dependent noise, we propose a truncated sequential procedure with a fixed mean-square accuracy. The asymptotic distribution of the estimator depends on the type of the noise distribution: it is normal when the noise has a Kotz’s distribution, while it is a mixture of normal distributions if the noise distribution is a variance mixture of normal distrbutions as well. In both cases, the convergence to the limiting distribution is uniform in the unknown parameter.      

A new method to detect periodically correlated structure

In this paper, we introduce a new method to test whether a discrete-time periodically correlated model explains an observed time series. The proposed method is based on the estimation of the support of spectral measure. Comparisons between our procedure and the methods which were proposed by Broszkiewicz-Suwaj et al. (Phys A 336:196–205, 2004) show that our testing procedure is more powerful. We investigate the performance of the proposed method by using real and simulated datasets.     

On first and second order stationarity of random coefficient models

We give conditions for first and second order stationarity of mixture autoregressive processes. We obtain a simple condition for positive definiteness of the solution of a generalisation of the Stein’s equation with semidefinite right-hand side and apply it to second order stationarity. The said condition may be of independent interest.     

Nonlinear autoregressive models with heavy-tailed innovation

In this paper, we discuss the relationship between the stationary marginal tail probability and the innovation's tail probability of nonlinear autoregressive models. We show that under certain conditions that ensure the stationarity and ergodicity, one dimension stationary marginal distribution has the heavy-tailed probability property with the same index as that of the innovation's tail probability.     

Bayesian analysis of conditional autoregressive models

Conditional autoregressive (CAR) models have been extensively used for the analysis of spatial data in diverse areas, such as demography, economy, epidemiology and geography, as models for both latent and observed variables. In the latter case, the most common inferential method has been maximum likelihood, and the Bayesian approach has not been used much. This work proposes default (automatic) Bayesian analyses of CAR models. Two versions of Jeffreys prior, the independence Jeffreys and Jeffreys-rule priors, are derived for the parameters of CAR models and properties of the priors and resulting posterior distributions are obtained. The two priors and their respective posteriors are compared based on simulated data. Also, frequentist properties of inferences based on maximum likelihood are compared with those based on the Jeffreys priors and the uniform prior. Finally, the proposed Bayesian analysis is illustrated by fitting a CAR model to a phosphate dataset from an archaeological region.     

Multiple autoregressive models with random coefficients

This paper derives conditions for the stationarity of a class of multiple autoregressive models with random coefficients. The models considered include as special cases those previously discussed by Andel (Ann. Math. Statist.42 (1971), 755–759; Math. Operationsforsch. Statist.7 (1976), 735–741).