Multistage sampling for Population Median Has there been any research done on how to estimate the population median using multistage sampling? Simple random sampling is not possible in my case I would really like to use something like multistage sampling.
 A: Suppose you are interested in the median $m$.  For the ordered  values, $x_1, x_2, \dots x_n$, compute the estimated weighted proportions $\hat{F}(x_1), \hat{F}(x_2) \dots$, where $\hat{F}(x_i)$ is the estimated proportion of observations $ X $ that are $ \le x_i$.
\begin{equation}
\hat{F}(x_i)= \frac{\sum_{j=1}^n w_j I(x_j\le x_i)}{\sum_{j=1}^n w_j}
\end{equation}
Here $w_j$ is the sampling weight for observation $j$.
Imagine that 37 and 41 are successive values of x in the sample such that  $\hat{F}(37) = 0.46$ and that $\hat{F}(41) = 0.53$. Since $F(m)=0.50$, it's obvious that the  estimated median $\hat{m}$  must be between 37 and 41. The value of $\hat{m}$ can be calculated by linear interpolation.
$$
\begin{aligned}
\hat{m} &=  37 + \frac{0.50 -0.46}{0.53 -0.46}\times(41-37) \\
& = 37 + \frac{.04}{.07}\times 4 = 39.28
\end{aligned}
$$
Note that this result depends only on the sampling weights, not on any other aspects of the survey design. The first reference that I know of is Woodruff (1952). Other quantiles are estimated in a similar way.
What has been the subject of research is the estimation of confidence intervals for quantiles. Woodruff (1952) contained a method that is based on the asymptotic normality of the sample weighted proportions $\hat{F}$. The method is still widely used. See the quick reference list below.
Some survey packages that estimate quantiles with standard errors/confidence intervals


*

*R: Tom Lumley's Survey package includes the svyquantile    function.

*Stata: Stas Kolenikov's epctile command (findit epctile in Stata)

*SUDAAN

*SAS SVYMEANS

*WesVar
References
Binder DA (1991) Use of estimating functions for interval estimation from complex surveys. Proceedings of the ASA Survey Research Methods Section 1991: 34-42
Dorfman A, Valliant R (1993) Quantile variance estimators in complex surveys. Proceedings of the ASA Survey Research Methods Section. 1993: 866-871
Francisco, C. A. and Fuller, W. A. 1991. “Quantile Estimation With a Complex Survey Design,”. The Annals of Statistics, 19: 454–469.
Shah BV, Vaish AK (2006) Confidence Intervals for Quantile Estimation from Complex Survey Data. Proceedings of the Section on Survey Research Methods.
http://www.amstat.org/sections/SRMS/Proceedings/y2012/files/304420_73075.pdf
Sitter, Randy R, and Changbao Wu. 2001. A note on Woodruff confidence intervals for quantiles. Statistics & probability letters 52, no. 4: 353-358.
Woodruff, Ralph S. 1952. Confidence intervals for medians and other position measures. Journal of the American Statistical Association 47, no. 260: 635-646.
