  Maximum-likelihood exponent maps have been studied as a technique to increase
the understanding and improve the fit of power-law exponents to experimental
and numerical simulation data, especially when they exhibit both upper and
lower cut-offs. The use of the technique is tested by analyzing seismological
data, acoustic emission data and avalanches in numerical simulations of the
3D-Random Field Ising model. In the different examples we discuss the nature of
the deviations observed in the exponent maps and some relevant conclusions are
drawn for the physics behind each phenomenon.
