Multiple change points tests against the umbrella alternative

We consider the problem of testing the null hypothesis of no change against the alternative of multiple umbrella-type change points in a series of independent observations. We extend the tests of Mack Wolfe (1981) to the change point set-up. We consider the two cases of known and unknown umbrella peak point. We obtain the asymptotic null distributions of the proposed tests and give approximations for their limiting critical values. We also give tables for their finite sample Monte Carlo critical values. We report the results of several Monte Carlo power studies conducted to compare the proposed tests with a number of multiple change points tests. As an illustration we applied the proposed tests using a real data set.