Alain Latour

Integer-Valued GARCH Process

An integer-valued analogue of the classical generalized autoregressive conditional heteroskedastic (GARCH) (p,q) model with Poisson deviates is proposed and a condition for the existence of such a process is given. For the case p = 1, q = 1, it is explicitly shown that an integer-valued GARCH process is a standard autoregressive moving average (1, 1) process. The problem of maximum likelihood estimation of parameters is treated. An application of the model to a real time series with a numerical example is given. Copyright 2006 The Authors Journal compilation 2006 Blackwell Publishing Ltd.

Existence and Stochastic Structure of a Non-negative Integer-valued Autoregressive Process

A simple criterion is given for the existence of a generalized integer-valued autoregressive (GINAR(p)) process. We show that the GINAR(p) process is nothing but an AR(p) process. The spectral density gives a good insight into the stochastic structure of a GINAR(p) model. The spectral representation of the process is explicitly given. The estimation of parameters of the process is also discussed and clarifies some results presented by Du and Li (The integer-valued autoregressive (INAR(p)) model. J. Times Ser. Anal., 12 (1991), 129--42). Finally, we describe the number of seizures of an epileptic patient using a model of this class.

THE MULTIVARIATE GINAR(p) PROCESS

A criterion is given for the existence of a stationary and causal multivariate integer-valued autoregressive process, MGINAR(p). The autocovariance function of this process being identical to the autocovariance function of a standard Gaussian MAR(p), we deduce that the MGINAR(p) process is nothing but a MAR(p) process. Consequently,, the spectral density is directly found and gives good insight into the stochastic structure of a MGINAR(p). The estimation of parameters of the models as well as the forecasting of the series, is discussed.

A simple integer-valued bilinear time series model

In this paper, we extend the integer-valued model class to give a nonnegative integer-valued bilinear process, denoted by INBL(p,q,m,n), similar to the real-valued bilinear model. We demonstrate the existence of this strictly stationary process and give an existence condition for it. The estimation problem is discussed in the context of a particular simple case. The method of moments is applied and the asymptotic joint distribution of the estimators is given: it turns out to be a normal distribution. We present numerical examples and applications of the model to real time series data on meningitis and Escherichia coli infections.

Inference on the Hurst Parameter and the Variance of Diffusions Driven by Fractional Brownian Motion

The use of diffusion models driven by fractional noise has become popular for more than two decades. The reasons that produced this situation have been varied in nature. We can mention, among others, those that come from mathematics and other from the applications. With respect to the first group, it should be noted that fractional Brownian motion (fBm) has interesting properties. First, it is self-similar. This property implies that such a process is, from the standpoint of its distribution, invariant with respect to scale transformations. Moreover the fractional noise, the process of increments of the fBm taking in a mesh of equally spaced points, satisfies a strong dependence condition that is a notion away from independence and mixing. Using this last property, it has been possible to model natural phenomena, which exhibit temporal correlations tending to zero so slowly that their sum tends to infinity. With regard to the applications, we should mention that fractional models have become popular for modeling real-life events such as the value assets in financial markets, models of chaos in quantum physics, rivers flow along the time, irregular images, weather events and contaminant transport problems, among others. The fBm is a mean zero Gaussian process with stationary increments and whose covariance function is uniquely determined by the Hursts parameter, that we denote by

Variance estimator for fractional diffusions with variance and drift depending on time

H

Estimation of the Parameters

and that is between zero and one. The value

On the behaviour of the sample autocovariances and autocorrelations of a seasonal arima model

H=\frac12

Simulation Algorithms and Simulation Studies

is important because the associate process results is the Brownian motion (Bm). The parameter

Some Pascal Procedures and Functions

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