Let X_1n,...,X_nn be independent random elements with an unknown change point θ∈(0,1), that is X_in has a distribution ν₁ or ν₂, respectively, according to i≤[nθ] or i>[nθ]. We propose an estimator θ_n of θ, which is defined as the maximizer of a weighted empirical process on (0,1). Finding upper bounds of polynomial and exponential type for the tails of nθ_n-[nθ], we are able to derive rates of almost sure convergence, of distributional convergence, of L_p-convergence and of convergence in the Ky-Fan- and in the Prokhorov-metric.