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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?

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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 $\mu$.

However, as pointed out at the end of the wiki article, exact computation of the LTS is impossible except for small data sets. This is why [0] introduced FastLTS, a fast algorithm to compute a consistent estimator of the exact LTS for moderately sized data set. You can find good implementations of FastLTS is many software packages. R has one (function robustbase::ltsReg).

  • [0] Rousseeuw, P. Van Driessen, K. (2006). Computing LTS Regression for Large Data Sets. Data Mining and Knowledge Discovery. vol:12 issue:1 pages:29-45.
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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.

Tukey's biweight is a commonly used function.

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