# Modeling trimmed mean

In OLS, the conditional mean $E(Y \mid X)$ is modeled as a function of some regressors $X$, i.e. $$E(Y \mid X) = X \beta.$$

Is there a regression technique that allows to model the conditional trimmed mean of $Y$?

(Least absolute deviations will lead to the extreme case of maximal trimming.)

Is there a direct relation between this question and least trimmed squares estimator?

Yes. The LTS (the wiki page you point to) is the regression equivalent of the trimmed mean: in LTS, one minimizes the sum of the smallest $h$ residuals to $\pmb x'\pmb\beta$, the regression hyper-plane, instead of the sum of the $h$ smallest residual to a fixed number $\mu$.
Look up robust statistics and M-estimators. Essentially, rather than assuming that $P(Y|X) \sim N[y-\beta X, \sigma^2]$, you choose some other function which downweights outliers. It's not quite a trimmed mean as such, but you get all the benefits likelihood methods, including standard errors on coefficients.