Median-unbiased estimators
Section of Estimation, part of Statistical inference
Median-unbiased estimators
| Type | Section |
| Taught in | Estimation |
| Part | Statistical inference |
An estimator T of a scalar \theta is median-unbiased when, for every \theta,
\mathbb{P}_\theta(T>\theta)=\mathbb{P}_\theta(T<\theta).
The median of the sampling distribution of T is \theta. This is a different impartiality from \mathbb{E} T=\theta. A skewed sampling distribution can be unbiased and still sit mostly on one side of \theta, or median-unbiased and still have a mean pulled by a long tail.
For a continuous distribution symmetric about \theta, the sample median is median-unbiased, because the number of observations above \theta is \operatorname{Bin}(n,1/2) and the sample median falls above \theta when more than half the sample does. The sample mean is unbiased, and under symmetry it is median-unbiased as well. The two properties part company for an asymmetric family. The uniform maximum M on (0,\theta) has \mathbb{E} M=n\theta/(n+1), so it is biased low. Its median solves \mathbb{P}(M\le m)=(m/\theta)^n=1/2, so the median is \theta\,2^{-1/n}, which is also below \theta. The rescaled estimator M\,2^{1/n} is median-unbiased. It is not the unbiased estimator (n+1)M/n.
Used in
- Taught in Estimation
- All 6 questions on Estimation
See also
Estimator · Method of moments · Sample correlation coefficient · Likelihood function · Maximum likelihood · Invariance of maximum likelihood · Unbiased estimator · Mean squared error