---
tag: PV187HG1K6
kind: definition
name: L-estimator
msc: [62G05, 62G30, 62G35]
defines: E3C4F0RKJY
uses: [58NHGZDSRE, 0TTNRP5PFQ]
sources:
  - url: https://en.wikipedia.org/wiki/L-estimator
---

## Definition

Let $X_1, \dots, X_n$ be real random variables, and let $X_{(1)} \le X_{(2)} \le \dots \le X_{(n)}$ be their [[order statistics|58NHGZDSRE]].
An *L-estimator* is an [[estimator|0TTNRP5PFQ]] of the form
$$T_n = \sum_{i=1}^{n} c_{n,i} \, X_{(i)},$$
where the coefficients $c_{n,1}, \dots, c_{n,n} \in \R$ are fixed numbers that do not depend on the data.

## Remarks

Common examples:

1. the sample mean, with $c_{n,i} = 1/n$;
2. the sample median, with $c_{n,(n+1)/2} = 1$ for odd $n$, and $c_{n,n/2} = c_{n,n/2+1} = 1/2$ for even $n$;
3. the $k$-trimmed mean, with $c_{n,i} = 1/(n - 2k)$ for $k < i \le n - k$ and $c_{n,i} = 0$ otherwise;
4. the midrange $(X_{(1)} + X_{(n)})/2$ and the range $X_{(n)} - X_{(1)}$.

Example: data $(4, 1, 100, 2, 3)$, sorted $(1, 2, 3, 4, 100)$.
The $1$-trimmed mean drops $1$ and $100$ and gives $(2 + 3 + 4)/3 = 3$, while the mean is $22$.

Non-example: the sample variance $\frac{1}{n-1}\sum_i (X_i - \bar X)^2$ is a quadratic function of the order statistics, not a linear one, so it is not an L-estimator.

Often the weights come from a function $J$ on $[0, 1]$ through $c_{n,i} = \frac1n J\big(\frac{i}{n+1}\big)$; then $T_n$ estimates $\int_0^1 J(u) F^{-1}(u) \, du$, where $F^{-1}$ is the quantile function.