Editing Median (section)

=== Estimation of variance from sample data ===

The value of <math>(2 f(x))^{-2}</math>—the asymptotic value of <math>n^{-1/2} (\nu - m)</math> where <math>\nu</math> is the population median—has been studied by several authors. The standard "delete one" [[Resampling (statistics)#Jackknife|jackknife]] method produces [[consistent estimator|inconsistent]] results.<ref name=Efron1982>{{cite book |last=Efron |first=B. |year=1982 |title=The Jackknife, the Bootstrap and other Resampling Plans |publisher=SIAM |location=Philadelphia |isbn=0898711797 }}</ref> An alternative—the "delete k" method—where <math>k</math> grows with the sample size has been shown to be asymptotically consistent.<ref name=Shao1989>{{cite journal |last1=Shao |first1=J. |last2=Wu |first2=C. F. |year=1989 |title=A General Theory for Jackknife Variance Estimation |journal=[[Annals of Statistics|Ann. Stat.]] |volume=17 |issue=3 |pages=1176–1197 |jstor=2241717 |doi=10.1214/aos/1176347263|doi-access=free }}</ref> This method may be computationally expensive for large data sets. A bootstrap estimate is known to be consistent,<ref name=Efron1979>{{cite journal |last=Efron |first=B. |year=1979 |title=Bootstrap Methods: Another Look at the Jackknife |journal=[[Annals of Statistics|Ann. Stat.]] |volume=7 |issue=1 |pages=1–26 |jstor=2958830 |doi=10.1214/aos/1176344552|doi-access=free }}</ref> but converges very slowly ([[computational complexity theory|order]] of <math>n^{-\frac{1}{4}}</math>).<ref name=Hall1988>{{cite journal |last1=Hall |first1=P. |last2=Martin |first2=M. A. |s2cid=119701556 |year=1988 |title=Exact Convergence Rate of Bootstrap Quantile Variance Estimator |journal=Probab Theory Related Fields |volume=80 |issue=2 |pages=261–268 |doi=10.1007/BF00356105 |doi-access=free }}</ref> Other methods have been proposed but their behavior may differ between large and small samples.<ref name=Jimenez-Gamero2004>{{cite journal |last1=Jiménez-Gamero |first1=M. D. |last2=Munoz-García |first2=J. |first3=R. |last3=Pino-Mejías |year=2004 |title=Reduced bootstrap for the median |journal=Statistica Sinica |volume=14 |issue=4 |pages=1179–1198 |url=http://www3.stat.sinica.edu.tw/statistica/password.asp?vol=14&num=4&art=11 }}</ref>