Non-Parametric Analyses of Log-Periodic Precursors to Financial Crashes
  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. 
Wei-Xing Zhou (UCLA) and Didier Sornette (UCLA and CNRS-Univ. Nice)