  We present a review of some recent results on estimation of location
parameter for several models of observations with cusp-type singularity at the
change point. We suppose that the cusp-type models fit better to the real
phenomena described usually by change point models. The list of models includes
Gaussian, inhomogeneous Poisson, ergodic diffusion processes, time series and
the classical case of i.i.d. observations. We describe the properties of the
maximum likelihood and Bayes estimators under some asymptotic assumptions. The
asymptotic efficiency of estimators are discussed as well and the results of
some numerical simulations are presented. We provide some heuristic arguments
which demonstrate the convergence of log-likelihood ratios in the models under
consideration to the fractional Brownian motion.
