Statistical Papers

, Volume 60, Issue 3, pp 717–746 | Cite as

Consistent nonparametric tests for detecting gradual changes in the marginals and the copula of multivariate time series

  • Jean-François QuessyEmail author
Regular Article


From a series of observations \({\mathbf Y}_1, \ldots , {\mathbf Y}_n\) in \(\mathbb {R}^d\) taken sequentially, an interesting question is to know whether or not a significant change occurred in their stochastic behavior. The problem has been largely investigated both for univariate and multivariate observations, where the null hypothesis states that \(F_1 = \cdots = F_n\), where \(F_j({\mathbf y}) = \mathrm{P}({\mathbf Y}_j \le {\mathbf y})\). In most of the works done so far, the alternative hypothesis is generally that of an abrupt change at some unknown time K, i.e. \(F_j = D_1\) for \(j \le K\) and \(F_j = D_2\) when \(j > K\). This assumption is unrealistic in applications where changes tend to occur gradually. In this paper, a more general gradual-change model is proposed in which one admits the existence of times \(K_1 < K_2\) where the distribution smoothly changes from \(D_1\) to \(D_2\). A general class of consistent test statistics for the detection of gradual changes is introduced and their large-sample behavior is investigated under a general \(\alpha \)-mixing condition. The proposed framework allows to detect changes in the marginal series as well as in the copula. Monte-Carlo simulations indicate the good sampling properties of the tests and their usefulness is illustrated on climatic data.


Copula Gradual-change model Sequential empirical process Serial multiplier bootstrap 



Two referees are gratefully acknowledged for their suggestions that led to an improvement of this work. Ph.D. student Félix Camirand Lemyre is also gratefully acknowledged for his help on the simulations and for suggesting the recursive formulas for the computation of the test statistics. This research was supported in part by an individual grant from the Natural Sciences and Engineering Research Council of Canada (NSERC) and by the Canadian Statistical Science Institute (CANSSI).


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Département de mathématiques et d’informatiqueUniversité du Québec à Trois-RivièresTrois-RivièresCanada

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