Ecosystems close to a critical threshold lose resilience, in the sense that perturbations can more easily push them into an alternative state. Recently, it has been proposed that such loss of resilience may be detected from elevated autocorrelation and variance in the fluctuations of the state of an ecosystem due to critical slowing down; the underlying generic phenomenon that occurs at critical thresholds. Here we explore the robustness of autocorrelation and variance as indicators of imminent critical transitions. We show both analytically and in simulations that variance may sometimes decrease close to a transition. This can happen when environmental factors fluctuate stochastically and the ecosystem becomes less sensitive to these factors near the threshold, or when critical slowing down reduces the ecosystem's capacity to follow high-frequency fluctuations in the environment. In addition, when available data is limited, variance can be systematically underestimated due to the prevalence of low frequencies close to a transition. By contrast, autocorrelation always increases toward critical transitions in our analyses. To exemplify this point, we provide cases of rising autocorrelation and increasing or decreasing variance in time series prior to past climate transitions.