Regression analysis with truncated samples and its application to ground-motion attenuation studies

The estimation of empirical attenuation laws with standard regression techniques commonly assumes the lognormal distribution of the response variable (e.g., peak ground acceleration [PGA]) for fixed values of the predictor variables (e.g., magnitude and distance from the source). Such an assumption may be invalidated by restrictions on the available sample induced by the acquisition system or imposed by the analyst, so that bias may be introduced in the estimation of regression parameters. In this article I analyze the distortion from lognormality due to station triggering. I propose a technique based on truncated regression analysis that does not require knowledge of which stations did not trigger. Furthermore, I introduce randomly truncated regression analysis to deal with thresholds that change randomly over time. This technique is adopted for stations that trigger based on the ratio between a short (STA) and long-term average (LTA) of the signal (STA/LTA ratio). I show the effectiveness of these techniques with applications to the estimation of PGA attenuation relations for both synthetic and real data sets. Real applications will refer to strong motion data from the European area and to weak motion data collected in the Friuli-Venezia Giulia and Veneto regions (northeast Italy).