## Multivariate changepoint problem.(English)Zbl 1029.62050

From the introduction: A simple multivariate changepoint problem can be formulated as follows. Let $${\mathbf X}_1,\dots,{\mathbf X}_{\mathbf N}$$ be a sequence of $$N$$ independent random vectors of dimension $$p$$, where $${\mathbf X}_{ \mathbf i}=(X_{1i},\dots,X_{pi})^t$$ for $$i=1,\dots,N$$. Let $$F({\mathbf X}_i,{\mathcal B }_i)$$ be the continuous distribution function (cdf) of the random vector $${\mathbf X}_{\mathbf i}$$ and where $${\mathcal B}_{\mathbf i}=({\mathcal B}_{1i},\dots,{\mathcal B}_{pi})^t$$ for $$i=1, \dots, N$$ are parameters. The above sequence of random vectors is said to have a changepoint at time point $$n$$ $$(1\leq n<N)$$, if the random vectors $${ \mathbf X}_{\mathbf i}$$ for $$i=1,\dots,n$$ have the cdf’s $$F({\mathbf X}_{\mathbf i}, \mathbf{0})$$ and the random vectors $${\mathbf X}_{\mathbf i}$$ for $$i=n+1,\dots,N$$ have the cdf’s $$F({\mathbf X}_{\mathbf i},{\mathcal B})$$ with $$\mathbf{0}=(0,\dots,0)^t$$ and $${\mathcal B}=({\mathcal B}_1,\dots, {\mathcal B}_p)^t$$. The time point $$n$$ may be called the single changepoint. The change may also occur smoothly over a period of time and the time point $$n$$ at which the change begins to occur may be called the continuous changepoint.
Studies about changepoint problems in a multivariate setting are very rarely found in the literature. We formulate two changepoint models, namely the single changepoint model (SC model) and the continuous changepoint model (CC model). Furthermore, we propose the appropriate test statistics for testing location, scale, and simultaneous location and scale changes in the above mentioned changepoint models. Section 2 contains the formulation of these models and the derivation of the appropriate test statistics. In Section 3, the asymptotic distributions of the proposed statistics are given and the concluding remarks are given in Section 4.

### MSC:

 62H15 Hypothesis testing in multivariate analysis 62E20 Asymptotic distribution theory in statistics 62H10 Multivariate distribution of statistics