  We point out that ignoring nonlinear effects in finite size scaling may lead
to errors in estimates of the critical temperature and Binder cumulants. We
show that the order of magnitude of these effects can be estimated from data at
relatively small volume. Using this estimate, we propose to use linear fits in
increasingly small temperature regions as the volume is increased (rather than
using a fixed temperature interval). The choice of the exact coefficient of
proportionality can be optimized and reveals interesting crossing patterns
among estimates. We show that the new procedure works very well for Dyson's
hierarchical model. We discuss applications of the method for 3 dimensional
spin models and finite temperature lattice gauge theories and comment on the
nonlinear effects for existing calculations.
