Censored regression methods for analyzing extreme end of a normally-distributed variable 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? 
 A: Could you clarify what you did when you said you "replaced all the values below the 90th percentile with the value corresponding to that percentile?"
That part is not clear to me, for one. I presume you mean that you discretized the continuous values to the left of your cutoff. It's not obvious, according to econometric or statistical theory, why you should choose to replace the data to the left of that point in the first place. 
Josh Angrist and Jorn-Steffen Pischke, in their book Mostly Harmless Econometrics, strongly advise against using censored data approaches when the data is not truly censored. For one, using approaches like Tobit require distributional assumptions, potential reductions in efficiency, and complicated functional forms to find marginal effects. See this discussion in Chapter 3, pages 94-102. There are other censored and zero-inflated approaches which do not require normality but still face the same criticisms I referenced above.
Hence, the recommendation is to use linear-regression and pay particular attention to the structure of your standard error. 
