Let Xin,..., Xnn, n є IN, be independent random elements with values in a measurable space. Suppose that for some θ є (0,1], X1n,..., X[nθ],n have distribution v1 and X[nθ]+1,n)..., Xnn have distribution v2≠v1, both unknown. We investigate an estimator θn for the change-point θ if actually no change has occured, i.e. θ=1 and all data Xin are i.i.d.. We prove that θn converges in law to the uniform distribution on (0,1). Furthermore, with probability one the sequence (θn)nєIN does not converge. The first result leads to a test of Kolmogorov-Smirnov type for the test problem H0:θ=1 versus H1:θ є (0,1). The test is baaed on θn and involves a resampling method. We show that it is an asymptotic level-α test, which is consistent on a large class of alternatives.