  We apply two non-parametric methods to test further the hypothesis that
log-periodicity characterizes the detrended price trajectory of large financial
indices prior to financial crashes or strong corrections. The analysis using
the so-called (H,q)-derivative is applied to seven time series ending with the
October 1987 crash, the October 1997 correction and the April 2000 crash of the
Dow Jones Industrial Average (DJIA), the Standard & Poor 500 and Nasdaq
indices. The Hilbert transform is applied to two detrended price time series in
terms of the ln(t_c-t) variable, where t_c is the time of the crash. Taking all
results together, we find strong evidence for a universal fundamental
log-frequency $f = 1.02 \pm 0.05$ corresponding to the scaling ratio $\lambda =
2.67 \pm 0.12$. These values are in very good agreement with those obtained in
past works with different parametric techniques.
