We consider the problems of confidence estimation and hypotheses testing of the parameter of a signal observed in a Gaussian white noise. For these problems, we point out lower bounds of asymptotic efficiency in the zone of moderate deviation probabilities. These lower bounds are versions of the local asymptotic minimax Hajek–Le Cam lower bound in estimation and the lower bound for the Pitman efficiency in hypotheses testing. The lower bounds are obtained both for logarithmic and sharp asymptotics of moderate deviation probabilities.