Summary We analyse Lagrange Multiplier (LM) tests for a shift in mean of a univariate time series at an unknown date. We consider a class of LM statistics based on non-parametric kernel estimators of the long-run variance and we develop a fixed-b asymptotic theory for the statistics. We provide results for the case of I(0) and I(1) errors. The fixed-b theory suggests that, for a given statistic, kernel and significance level, there exists a bandwidth such that the fixed-b asymptotic critical value is the same for both I(0) and I(1) errors. We compute these ‘robust’ bandwidths for a selection of well-known kernels. In the case of the supremum statistic, the robust bandwidth LM tests have good power that is monotonic and similar for all kernels. In the case of the mean and exponential mean statistics, power is non-monotonic when the robust bandwidths are used. Under the null hypothesis, the Bartlett supremum statistic has little, if any, over-rejection problems in finite samples even when there is a moving average component with a negative coefficient. In contrast, the other supremum statistics tend to over-reject in this case. In practice, we recommend the Bartlett supremum LM statistic implemented with the robust bandwidth.