We consider a class of simple estimators of the change-point m in a sequence of n independent random variables X₁…,X_n satisfying E(X_i) = θ₀ for i = 1,…,m and E(X_i) = θ₀+δ_n for i = m +1,…n. (θ₀ and δ_n are unknown). We obtain rates of consistency for the estimator, derive its limiting distribution and show that the bootstrap approximation is asymptotically valid. The results are illustrated by some simulations.