Implements an amazing theoretical result for the median confidence interval
which is valid for any distribution, and only assumes iid for the samples.
This result involves the binomial distribution.
It is described here on these webpages:
perfeval.epfl.ch/printMe/conf-1.ppt (see p. 12)
http://www.stat.ufl.edu/STA6166/Fall06/17.STA6166%20Chapter5p2.pdf
http://www.behav.org/QP/quant_large.pdf (see Appendix II)
http://stat-www.berkeley.edu/~stark/Teach/S240/Notes/ch5.htm (see the final section)
http://www-users.york.ac.uk/~mb55/intro/cicent.htm
http://www.statsdirect.com/help/nonparametric_methods/qci.htm
and in these books:
Nonparametrics: Statistical Methods Based on Ranks, Holden-Day, 1975, by Lehmann p.182-183
Introduction to Mathematical Statistics (6th Edition)
by Robert V. Hogg, Allen Craig, by Joseph W. McKean (~p. 246 has an interesting discussion of this; see also their discussion in chapter 10)
Practical Nonparametric Statistics (3rd edition), Wiley 1999, by Conover WJ (several people on the web referenced this)
http://books.google.com/books?id=tARVZq4hq7UC&pg=PA173&lpg=PA173&dq=binomial+distribution+cdf+confidence+interval+median&source=web&ots=yFfj2N9vFi&sig=IilUl90CEaCqul8GCgpEASRjKVY#PPA173,M1
If you do not want to calculate it, here are tables:
http://www.math.unb.ca/~knight/utility/MedInt95.htm
http://biomet.oxfordjournals.org/cgi/content/abstract/57/3/613
and here is an applet:
http://www.sph.emory.edu/~cdckms/median-final.html
The sole drawback of this binomial result is that it is only exact for certain discrete values of the confidence level.
In particular, the exact confidence levels are limited to a small set of values
that is a function of the number of samples and the binomial distribution's cdf.
To work around this limitation, 2 solutions are possible.
First, if the confidence level that you desire is not one of the exact values,
you may simply use the confidence interval produced by the next largest exact confidence level.
This will be a conservative (too large) confidence interval for the level that you desire,
but for large sample sizes, the error should typically be small.
Second, a more rigorous technique is to use some form of interpolation.
For example, the Hettmansperger-Sheather result uses linear interpolation to handle arbitrary confidence levels.
This technique was originally described here:
Confidence Interval Based on Interpolated Order Statistics,
Hettmansperger, T. P., and Sheather, S. J.
Statistical Probability Letters, 4: 75–79 1986
and some web references are:
http://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/mediancl.htm
support.spss.com/Student/Documentation/Algorithms/14.0/errorbars.pdf
This article claims that linear interpolation is the best that can do:
Interpolated Nonparametric Prediction Intervals and Confidence Intervals
Rudolf Beran, Peter Hall
Journal of the Royal Statistical Society. Series B (Methodological), Vol. 55, No. 3 (1993), pp. 643-652
http://links.jstor.org/sici?sici=0035-9246%281993%2955%3A3%3C643%3AINPIAC%3E2.0.CO%3B2-W&size=LARGE&origin=JSTOR-enlargePage
and this one also advocates it:
Nonparametric Ranked-set Sampling Confidence Intervals for Quantiles of a Finite Population
Jayant V. Deshpande, Jesse Frey and Omer Ozturk
http://www.springerlink.com/content/h067ml43p62r8145/
See also this article:
http://www.springerlink.com/content/t501k266761l5461/
WARNING: if use these interpolation techniques, then the generality of the result is slightly reduced
(there IS now some distributional dependence):
"Confidence intervals for the population median based on interpolating adjacent order statistics are presented. They are shown to depend only slightly on the underlying distribution. A simple, nonlinear interpolation formula is given which works well for a broad collection of underlying distributions."
http://stinet.dtic.mil/oai/oai?verb=getRecord&metadataPrefix=html&identifier=ADA152607
In addition to the above binomial result, there is:
"... a method given by Wilcox (see Reference below) on page 87, is based on the Maritz-Jarrett estimate of the standard error for a quantile"
http://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/mediancl.htm
a better way than the binomial or bootstrap using smoothing:
http://www.informaworld.com/smpp/content~content=a780357685~db=all~jumptype=rss
this old paper; gives results for point and interval of mean, median and mode of log normal:
http://links.jstor.org/sici?sici=0006-3444%28197912%2966%3A3%3C567%3ACEOMOL%3E2.0.CO%3B2-W&size=LARGE&origin=JSTOR-enlargePage
a novel way of using median CIs to obtain robust estimates:
http://web.informatik.uni-bonn.de/IV/strelen/Lehre/Veranstaltungen/sim/Folien/24ASTC.pdf
- Specified by:
getMedianEst
in interface Bootstrap.UnitTest.Distribution