Editing Median (section)

==Median-unbiased estimators==
{{main|Bias of an estimator#Median-unbiased estimators}}
Any [[Bias of an estimator|''mean''-unbiased estimator]] minimizes the [[risk]] ([[expected loss]]) with respect to the squared-error [[loss function]], as observed by [[Gauss]]. A [[Bias of an estimator#Median unbiased estimators, and bias with respect to other loss functions|''median''-unbiased estimator]] minimizes the risk with respect to the [[Absolute deviation|absolute-deviation]] loss function, as observed by [[Laplace]]. Other [[loss functions]] are used in [[statistical theory]], particularly in [[robust statistics]].

The theory of median-unbiased estimators was revived by George W. Brown in 1947:<ref name="Brown" />

{{Blockquote|An estimate of a one-dimensional parameter θ will be said to be median-unbiased if, for fixed θ, the median of the distribution of the estimate is at the value θ; i.e., the estimate underestimates just as often as it overestimates. This requirement seems for most purposes to accomplish as much as the mean-unbiased requirement and has the additional property that it is invariant under one-to-one transformation.|page 584}}

Further properties of median-unbiased estimators have been reported.<ref name="Lehmann" /><ref name="Birnbaum" /><ref name="vdW" /><ref name="Pf" /> 

There are methods of constructing median-unbiased estimators that are optimal (in a sense analogous to the minimum-variance property for mean-unbiased estimators). Such constructions exist for probability distributions having [[monotone likelihood ratio|monotone likelihood-functions]].<ref>Pfanzagl, Johann. "On optimal median unbiased estimators in the presence of nuisance parameters." The Annals of Statistics (1979): 187–193.</ref><ref>{{cite journal | last1 = Brown | first1 = L. D. | last2 = Cohen | first2 = Arthur | last3 = Strawderman | first3 = W. E. | year = 1976 | title = A Complete Class Theorem for Strict Monotone Likelihood Ratio With Applications | url = http://projecteuclid.org/euclid.aos/1176343543 | journal = Ann. Statist. | volume = 4 | issue = 4| pages = 712–722 | doi = 10.1214/aos/1176343543 | doi-access = free }}</ref> One such procedure is an analogue of the [[Rao–Blackwell theorem|Rao–Blackwell procedure]] for mean-unbiased estimators: The procedure holds for a smaller class of probability distributions than does the Rao—Blackwell procedure but for a larger class of [[loss function]]s.<ref>{{cite journal | last1 = Page | last2 = Brown | first2 = L. D. | last3 = Cohen | first3 = Arthur | last4 = Strawderman | first4 = W. E. | year = 1976 | title = A Complete Class Theorem for Strict Monotone Likelihood Ratio With Applications | url = http://projecteuclid.org/euclid.aos/1176343543 | journal = Ann. Statist. | volume = 4 | issue = 4| pages = 712–722 | doi = 10.1214/aos/1176343543 | doi-access = free }}</ref>