lestimation

L-estimation, or L-estimation of location, is a class of statistical estimators defined as linear combinations of the sample’s order statistics. Given a sample X1, X2, ..., Xn and the order statistics X(1) ≤ X(2) ≤ ... ≤ X(n), an L-estimator of location has the form L = sum over i of w_i times X(i), where the weights w_i are fixed (often depending on n) and sum to 1. The estimator’s robustness is influenced by how the weights treat extreme observations.

The class includes common estimators such as the sample median, which can be viewed as selecting the central order statistic, and trimmed means, which assign zero weight to the extreme observations and equal weights to the middle portion. Other L-estimators use different weight schemes to balance robustness against efficiency. In practice, the choice of weights determines the estimator’s sensitivity to outliers and its performance under various data-generating scenarios.

Properties: L-estimators are straightforward to compute and can be tuned for robustness via the weight vector. They often exhibit asymptotic normality under regularity conditions, with the achieved efficiency depending on the chosen weights. The robustness of an L-estimator is described by its influence function and breakdown point, which improve when extreme order statistics receive less weight.

Applications: They are primarily used for estimating a univariate location parameter in the presence of contaminated data or outliers. While focused on location, the idea of linear combinations of order statistics extends conceptually to other estimation problems.

See also: M-estimation, R-estimation, order statistics, trimmed mean, median, robustness in statistics.