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Consider iid samples from a fixed distribution function $F(x)$ and consider its median. Now consider another median from iid samples where one half is drawn from $F_1(x)$ and the other is drawn from $F_2(x)$ with $F_1(x)+F_2(x)=2F(x)$ for all $x$. Which median has a larger variance?

lance
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  • Are you assuming that the sample size is always even? –  May 05 '12 at 09:13
  • Do you know how the variance of the median depends on the size of the sample? – leonbloy May 05 '12 at 12:14
  • Let us assume that they have the same number of even samples. – lance May 05 '12 at 12:58
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    Homework? If so, please add the homework tag... consider constructing an example, e.g., with a distribution on $\{-1,1\}$ to help develop some intuition. – jbowman May 05 '12 at 13:56
  • I don't think there is enough information about F1 and F2 to answer the question. Let m be the median of F then F(m)=1/2. Since F1(m)+F2(m)=2F(m), one of the Fis has a median less than m and the other has a median greater than m (assuming both F1(m) and F2(m) are greater than 0). But this doesn't even tell us which one has the larger median. The sample median is an order statistic and the properties of order statistics that can be found in David's book on order statistics and many nonparametric statistics tests may help. – Michael R. Chernick May 05 '12 at 14:12
  • Also Kendall and Stuart Advanced Theory of Statistics Volume 1 may have a lot of information on this. In fact I think there is a closed form solution for the variance of the median. But even given the formula I think you will find there is something missing to compute and compare the variances. My first guess is that the one with the larger median has the larger variance. I would check the formula though. This may be a trick question and F1 and F2 have the same variance for the sample median. That may mean that you do have sufficient information to solve the problem. – Michael R. Chernick May 05 '12 at 14:19
  • Yes I have seen Sen (1968) formula (2.9) that seems to suggest that variance is decreasing. Please advise. http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aoms/1177698155 – lance May 05 '12 at 16:15
  • I try to understand the relationship between this formula and http://stats.stackexchange.com/questions/16608/what-is-the-variance-of-the-weighted-mixture-of-two-gaussians – lance May 05 '12 at 17:43

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I did a Google search for variance of the sample median and found the 1978 paper by Maritz and Jarrett. I remember this paper as it is referenced in my bootstrap book where I used it to show an example where the bootstrap distribution can be obtained without need for the Monte Carlo approximation. I didn't go to the trouble of working out the solution to this problem based on the paper because this is a homework type problem and the site suggests providing hints to the solution rather than doing it for the author of the question. The paper is in JASA 1978.

Michael R. Chernick
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    The paper [On the median of a finite mixture](http://dx.doi.org/10.1080/00949659708811826) by Al-hussinp and Osman (Journal of Statistical Computation and Simulation 58(2) 1997 121--144) might (or might not) be helpful. It contains an iterative scheme for finding the median of a mixture. – მამუკა ჯიბლაძე Feb 10 '17 at 10:03