Optimizing regression coefficients to predict the largest outcomes What is a sound methodology to improve the efficiency of the regression coefficients when we are interested in predicting the larger values of the marginal distribution (tails)? 
For example, we want to predict seismic waves based on a number of covariates recorded by our probes. Data can not be assumed to be strictly linear, given that most earthquakes develop abruptly after a certain covariates threshold is reached. The vast majority of observations are not considered harmful and should be down-weighted in our analysis. What we are really interested in estimating are the more extreme outcomes.
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Weighted least square comes to mind, but how should the weights be calculated? 
Is quantile regression with, say, $\tau = [0.2, 0,8]$ a better approach?
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 A: It seems that you either need to relax linearity assumptions for covariates (e.g., use restricted cubic splines, i.e., natural splines) or to define a new optimality criterion (other than maximum likelihood) and then to optimize the fit to that.    I'm glad you clarified that you don't want to discard observations.  The optimality criterion will define the effective weights; you don't have to.
Least squares will give automatic emphasis to fitting more extreme $Y$ values at the expense of higher mean absolute error or median absolute error.
A: What you have written suggests that logistic regression, with attention to diagnostics and residuals, will be your best bet.  It can help you take into account nonlinearity in the relationship between predictor and outcome  You'll want to test multiplicative effects (interactions), as the thresholds you talk about may be joint thresholds involving multiple predictors.  I am concerned, though, by your statement that "The vast majority of observations are not considered harmful and will be discarded from our analysis."  In order to explain what causes the events of interest, it is absolutely necessary to know something about the conditions that do not produce it, just as it is to know about the ones that do.
