Joseph Tadjuidje Kamgaing

Changepoints in Times Series of Counts

In this article, we discuss the problem of testing for a changepoint in the structure of an integer‐valued time series. In particular, we consider a test statistic of cumulative sum type for general Poisson autoregressions of order 1. We investigate the asymptotic behaviour of conditional least‐squares estimates of the parameters in the presence of a changepoint. Then, we derive the asymptotic distribution of the test statistic under the hypothesis of no change, allowing for the calculation of critical values. We prove consistency of the test, that is, asymptotic power 1, and consistency of the corresponding changepoint estimate. As an application, we have a look at changepoint detection in daily epileptic seizure counts from a clinical study.

On Geometric Ergodicity of CHARME Models

In this article we consider a CHARME model, a class of generalized mixture of nonlinear nonparametric AR-ARCH time series. To provide sets of conditions under which such processes are geometrically ergodic and, therefore, satisfy some mixing conditions, we apply the theory of Markov chains to derive asymptotic stability of this model. These results form the basis for deriving an asymptotic theory for nonparametric estimation. As an illustration, neural network sieve estimates for the autoregressive and volatility functions are considered, and consistency of the parameter estimates is obtained.

Testing for parameter stability in nonlinear autoregressive models

In this article we develop testing procedures for the detection of structural changes in nonlinear autoregressive processes. For the detection procedure, we model the regression function by a single layer feedforward neural network. We show that CUSUM‐type tests based on cumulative sums of estimated residuals, that have been intensively studied for linear regression, can be extended to this case. The limit distribution under the null hypothesis is obtained, which is needed to construct asymptotic tests. For a large class of alternatives, it is shown that the tests have asymptotic power one. In this case, we obtain a consistent change‐point estimator which is related to the test statistics. Power and size are further investigated in a small simulation study with a particular emphasis on situations where the model is misspecified, i.e. the data is not generated by a neural network but some other regression function. As illustration, an application on the Nile data set as well as S&P log‐returns is given.

Autoregressive Processes with Data-Driven Regime Switching

We develop a switching-regime vector autoregressive model in which changes in regimes are governed by an underlying Markov process. In contrast to the typical hidden Markov approach, we allow the transition probabilities of the underlying Markov process to depend on past values of the time series and exogenous variables. Such processes have potential applications in finance and neuroscience. In the latter, the brain activity at time t (measured by electroencephalograms) will be modelled as a function of both its past values as well as exogenous variables (such as visual or somatosensory stimuli). In this article, we establish stationarity, geometric ergodicity and existence of moments for these processes under suitable conditions on the parameters of the model. Such properties are important for understanding the stability properties of the model as well as for deriving the asymptotic behaviour of various statistics and model parameter estimators. Copyright 2009 Blackwell Publishing Ltd

Shrinkage Estimation for Multivariate Hidden Markov Models

ABSTRACT Motivated from a changing market environment over time, we consider high-dimensional data such as financial returns, generated by a hidden Markov model that allows for switching between different regimes or states. To get more stable estimates of the covariance matrices of the different states, potentially driven by a number of observations that are small compared to the dimension, we modify the expectation–maximization (EM) algorithm so that it yields the shrinkage estimators for the covariance matrices. The final algorithm turns out to reproduce better estimates not only for the covariance matrices but also for the transition matrix. It results into a more stable and reliable filter that allows for reconstructing the values of the hidden Markov chain. In addition to a simulation study performed in this article, we also present a series of theoretical results that include dimensionality asymptotics and provide the motivation for certain techniques used in the algorithm. Supplementary materials for this article are available online.

A uniform central limit theorem for neural network based autoregressive processes with applications to change-point analysis

We consider an autoregressive process with a nonlinear regression function that is modelled by a feedforward neural network. First, we derive a uniform central limit theorem which is useful in the context of change-point analysis. Then, we propose a test for a change in the autoregression function which – by the uniform central limit theorem – has asymptotic power one for a large class of alternatives including local alternatives not restricted to the correctly specified model.