  We address the characterization of the gain parameter of a non-deterministic
noiseless linear amplifier (NLA) and compare the performances of different
estimation strategies using tools from quantum estimation theory. At first, we
show that, contrary to naive expectations, post-selecting only the amplified
states does not offer the most accurate estimate. We then focus on minimal
implementations of a NLA, i.e. those obtained by coupling the input state to a
two-level system, and show that the maximal amount of information about the
gain of the NLA is obtained by measuring the whole composite system. The
quantum Fisher information (QFI) of this best-case scenario is analysed in some
details, and compared to the QFI of the post-selected states, both for
successful and unsuccessful amplification. Eventually, we show that full
extraction of the available information is achieved when the non-deterministic
process is implemented by a L\"uders instrument. We also analyse the precision
attainable by probing NLAs by single-mode pure states and measuring the field
or the number of quanta, and discuss in some details the specific cases of
squeezed vacuum and coherent states.
