  In a wide range of applications, the stochastic properties of the observed
time series change over time. The changes often occur gradually rather than
abruptly: the properties are (approximately) constant for some time and then
slowly start to change. In many cases, it is of interest to locate the time
point where the properties start to vary. In contrast to the analysis of abrupt
changes, methods for detecting smooth or gradual change points are less
developed and often require strong parametric assumptions. In this paper, we
develop a fully nonparametric method to estimate a smooth change point in a
locally stationary framework. We set up a general procedure which allows us to
deal with a wide variety of stochastic properties including the mean,
(auto)covariances and higher moments. The theoretical part of the paper
establishes the convergence rate of the new estimator. In addition, we examine
its finite sample performance by means of a simulation study and illustrate the
methodology by two applications to financial return data.
