This paper raises the question of adequacy of Anscombe-Chow-Robbins (ACR) type stopping rules in the inference for regular non-ergodic processes. It is shown that the usual recipe for constructing an ACR type stopping rule fails to provide an asymptotically risk efficient procedure for estimating the mean of a single-type supercitical branching process. However, the method does yield an ansymptotically efficient sampling rule in the sense of Chow and Robbins (1965). This is established using tools from large deviation theory for branching processes. These results illustrate that while the ACR rules may perform well for stationary and ergodic processes, they could fail in non-ergodic models.