U-Statistics for Change under Alternatives

Asymptotic distributions of U-statistics to test for possible changes in the distribution will be derived when the change occurred. We will show that for all possible types of kernels, symmetric, antisymmetric, degenerate, non-degenerate, the test statistics are asymptotically normally distributed. We also study the distribution of the estimator of the time of change. Its large sample behaviour is approximately that of the maximum of a two-sided random walk. The terms in these random walks explain the exact nature of bias in the change-point estimator. Several examples will be given as illustration.