I have a normally distributed continuous variable referring to an observed human behavior, and I'm interested in measuring or rather analyzing the extreme of this behavior, namely, the top 10% of the distribution as displayed in this graph. So I went ahead and replaced all the values below the 90th percentile with the value corresponding to that percentile (cutoff). This new variable is now exponentially-distributed and left-censored with a huge pile of data on the left. My goal is to conduct a multiple regression analysis with the censored variable as the DV.
Clarification: The reason why I replaced values with a cutoff is that there is no variation below the cutoff that is meaningful for the purposes of measuring the extreme behavior. For example, if the range of values is 1 to 40, and the cutoff is 20, I'm assuming that any value below 20 is not meaningfully different from 20 (and I have good reason to assume this).
Question: Someone suggested that I use tobit regression, but from what I've read tobit regression assumes that the censored data is normally distributed, whereas, in this case, I am no longer interested in the original variable--the new variable of interest represents an extreme behavior that is not normally distributed. If I am correct that tobit regression would not be appropriate, what would be an appropriate regression method to use?