We consider a nonparametric CUSUM test for change in the mean of multivariate time series with time varying covariance. We prove that under the null, the test statistic has a Kolmogorov limiting distribution. The asymptotic consistency of the test against a large class of alternatives which contains abrupt, smooth and continuous changes is established. We also perform a simulation study to analyze the size distortion and the power of the proposed test.

In the statistical literature there is a vast amount of works on testing for change in the mean of univariate time series. Sen and Srivastava [

In order to construct the test statistic let

The CUSUM test statistic we will consider is given by

The sequence of matrices

There exists

Suppose that Assumptions

Moreover, the cumulative distribution function of

To prove Theorem

For two random vectors

Consider an i.i.d. sequence

There are two sufficient conditions to prove that

the finite-dimensional distributions of

Assume that

Write

Then

For

We have

In order to prove Theorem

Assume that

Let

Then

Lemma 2 of Lai and Wei [

Under the null

We assume that under the alternative

There exists a function

There exists

There exists

Suppose that Assumptions

We have

Straightforward computation leads to

Without loss of generality we assume that under the alternative hypothesis

Suppose that Assumptions

It is easy to show that (

The result of Corollary

In this subsection we assume that the break in the mean does not happen suddenly but the transition from one value to another is continuous with slow variation. A well-known dynamic is the smooth threshold model (see Teräsvirta [

Two choices for the function

if

if

This means that at the beginning of the sample

Suppose that Assumptions

The assumptions (

Since

Assume that

In this subsection we will examine the behaviour

Suppose that Assumptions

The assumptions H

All models are driven from an i.i.d. sequences

In order to evaluate the size distortion of the test statistic

From Table

Empirical sizes (%).

1% | 0.2 | 0.3 | 0.4 | |

Model 1 | 5% | 2.1 | 2.9 | 4 |

10% | 4.9 | 7.3 | 8.9 | |

1% | 0.0 | 0.2 | 0.3 | |

Model 2 | 5% | 1.1 | 2.7 | 3.0 |

10% | 2.9 | 6.4 | 7.3 |

In order to see the power of the test statistic

In this model the mean and the covariance are subject to an abrupt change at the same time:

The mean is subject to an abrupt change and the covariance is time varying (see Figure

The three kinds of change in the mean.

We consider a logistic smooth transition for the mean and a time varying covariance (see Figure

In this model the mean is a polynomial of order two and the covariance matrix is also time varying as in the preceding Models

From Table

Empirical powers (%).

1% | 11.4 | 81.8 | 100 | |

Model 3 | 5% | 34.1 | 92.1 | 100 |

10% | 46.9 | 95.1 | 100 | |

1% | 3.9 | 49.8 | 99.9 | |

Model 4 | 5% | 15.7 | 71.4 | 99.9 |

10% | 29 | 80.4 | 100 | |

1% | 1.4 | 22.7 | 95.9 | |

Model 5 | 5% | 8.3 | 44.7 | 98.4 |

10% | 15.9 | 56.5 | 99.3 | |

1% | 1.4 | 17.2 | 94 | |

Model 6 | 5% | 8.5 | 38.5 | 97.7 |

10% | 16.3 | 51.9 | 98.7 | |

1% | 0.1 | 5.3 | 44.2 | |

Model 7 | 5% | 2.2 | 16.1 | 70.0 |

10% | 5.9 | 25.5 | 79.4 |

The author would like to thank the anonymous referees for their constructive comments.