This paper is devoted to the off-line multiple changepoint detection in a semiparametric framework. The time series is supposed to belong to a large class of models including AR(∞), ARCH(∞), TARCH(∞),... models where the coefficients change at each instant of breaks. The different unknown parameters (number of changes, change dates and parameters of successive models) are estimated using a penalized contrast built on conditional quasilikelihood. Under Lipshitzian conditions on the model, the consistency of the estimator is proved when the moment order r of the process satisfies r ≥ 2. If r ≥ 4, the same convergence rates for the estimators than in the case of independent random variables are obtained. The particular cases of AR(∞), ARCH(∞) and TARCH(∞) show that our method notably improves the existing results. 1. Introduction. The problem of the detection of change-points is a classical problem as well as in the statistic than in the signal processing community. If the first important result in this topic was obtained by Page [20] in 1955, real advances have been done in the seventies, notably with the results of Hinkley (see for instance Hinkley [13]) and the topic of change detection became a distinct and important field of the statistic since the eighties (see the book of Basseville and Nikiforov [3] for a large overview). Two approaches are generally considered for solving a problem of change detection: an 'on-line' approach leading to sequential estimation and an 'off-line' approach which arises when the series of observations is complete. Concerning this last approach, numerous results were obtained for independent random variables in a parametric frame (see for instance Bai and Perron [1]). The case of the off-line detection of multiple change-points in a parametric or semiparametric frame for dependent variables or time series also provided an important literature. The present paper is a new contribution to this problem. In this paper, we consider a general class M T (M, f ) of causal (non-anticipative) time series. Let M and f be a measurable functions such that for all (x i ) i∈IN ∈ IR IN , M (x i ) i∈IN is a (m × p) non-zero real matrix and f (x i ) i∈IN ∈ IR m . Let T ⊂ Z and (ξ t ) t∈Z be a sequence of centered independent and identically distributed (iid) R p -random vectors called the innovations and satisfying AMS 2000 subject classifications: Primary 62M10, 62F12
We consider here together the inference questions and the change-point problem in a large class of Poisson autoregressive models (see Tjøstheim, 2012 [34]). The conditional mean (or intensity) of the process is involved as a non-linear function of it past values and the past observations. Under Lipschitz-type conditions, it can be written as a function of lagged observations. For the latter model, assume that the link function depends on an unknown parameter θ 0 . The consistency and the asymptotic normality of the maximum likelihood estimator of the parameter are proved. These results are used to study change-point problem in the parameter θ 0 . From the likelihood of the observations, two tests are proposed. Under the null hypothesis (i.e. no change), each of these tests statistics converges to an explicit distribution. Consistencies under alternatives are proved for both tests. Simulation results show how those procedures work in practice, and applications to real data are also processed.
This paper studies the model selection problem in a large class of causal time series models, which includes both the ARMA or AR(∞) processes, as well as the GARCH or ARCH(∞), APARCH, ARMA-GARCH and many others processes. To tackle this issue, we consider a penalized contrast based on the quasi-likelihood of the model. We provide sufficient conditions for the penalty term to ensure the consistency of the proposed procedure as well as the consistency and the asymptotic normality of the quasi-maximum likelihood estimator of the chosen model. We also propose a tool for diagnosing the goodness-of-fit of the chosen model based on a Portmanteau test. Monte-Carlo experiments and numerical applications on illustrative examples are performed to highlight the obtained asymptotic results. Moreover, using a data-driven choice of the penalty, they show the practical efficiency of this new model selection procedure and Portemanteau test.
: We consider a process X = (X t ) t∈Z belonging to a large class of causal models including AR(∞), ARCH(∞), TARCH(∞),... models. We assume that the model depends on a parameter θ 0 ∈ IR d and consider the problem of testing for change in the parameter. Two statistics Q(1) n and Qn are constructed using quasi-likelihood estimator (QLME) of the parameter. Under the null hypothesis that there is no change, it is shown that each of these two statistics weakly converges to the supremum of the sum of the squares of independent Brownian bridges. Under the local alternative that there is one change, we show that the test statistic Q n = max Q (1) n , Q (2) n diverges to infinity. Some simulation results for AR(1), ARCH(1) and GARCH(1,1) models are reported to show the applicability and the performance of our procedure with comparisons to some other approaches.
: We propose a new sequential procedure to detect change in the parameters of a process X = (X t ) t∈Z belonging to a large class of causal models (such as AR(∞), ARCH(∞), TARCH(∞), ARMA-GARCH processes). The procedure is based on a difference between the historical parameter estimator and the updated parameter estimator, where both these estimators are based on a quasi-likelihood of the model. Unlike classical recursive fluctuation test, the updated estimator is computed without the historical observations. The asymptotic behavior of the test is studied and the consistency in power as well as an upper bound of the detection delay are obtained. Some simulation results are reported with comparisons to some other existing procedures exhibiting the accuracy of our new procedure. The procedure is also applied to the daily closing values of the Nikkei 225, S&P 500 and FTSE 100 stock index. We show in this real-data applications how the procedure can be used to solve off-line multiple breaks detection.
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