On nonparametric change point estimator based on empirical characteristic functions

Abstract

We propose a nonparametric change point estimator in the distributions of a sequence of independent observations in terms of the test statistics given by Hu\v{s}kova and Meintanis (2006) that are based on weighted empirical characteristic functions. The weight function $\omega(t; a)$ under consideration includes the two weight functions from Hu\v{s}kova and Meintanis (2006) plus the weight function used by Matteson and James (2014), where $a$ is a tuning parameter. Under the local alternative hypothesis, we establish the consistency, convergence rate, and asymptotic distribution of this change point estimator which is the maxima of a two-side Brownian motion with a drift. Since the performance of the change point estimator depends on $a$ in use, we thus propose an algorithm for choosing an appropriate value of $a$, denoted by $a_{\text{s}}$ which is also justified. Our simulation study shows that the change point estimate obtained by using $a_{\text{s}}$ has a satisfactory performance. We also apply our method to a real dataset.