  We numerically investigate the sensitivity to initial conditions of
asymmetric unimodal maps $x_{t+1} = 1-a|x_t|^{z_i}$ ($i=1,2$ correspond to
$x_t>0$ and $x_t<0$ respectively, $z_i >1$, $0<a\leq 2$, $t=0,1,2,...$) at the
edge of chaos. We employ three distinct algorithms to characterize the
power-law sensitivity to initial conditions at the edge of chaos, namely:
direct measure of the divergence of initially nearby trajectories, the
computation of the rate of increase of generalized nonextensive entropies $S_q$
and multifractal analysis. The first two methods provide consistent estimates
for the exponent governing the power-law sensitivity. In addition to this, we
verify that the multifractal analysis does not provide precise estimates of the
singularity spectrum $f(\alpha)$, specially near its extremal points. Such
feature prevents to perform a fine check of the accuracy of the scaling
relation between $f(\alpha)$ and the entropic index $q$, thus restricting the
applicability of the multifractal analysis for studing the sensitivity to
initial conditions in this class of asymmetric maps.
