We present an optimal strategy having finite outcomes for estimating a sing
le parameter of the displacement operator on an arbitrary finite-dimensiona
l system using a finite number of identical samples. Assuming the uniform a
priori distribution for the displacement parameter, an optimal strategy ca
n be constructed by making the square root measurement based on uniformly d
istributed sample points. This type of measurement automatically ensures th
e global maximality of the figure of merit, that is, the so-called average
score or fidelity. Quantum circuit implementations for the optimal strategi
es are provided in the case of a two-dimensional system.