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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?

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Are you assuming that the sample size is always even? – user10525 May 5 '12 at 9:13
Do you know how the variance of the median depends on the size of the sample? – leonbloy May 5 '12 at 12:14
Let us assume that they have the same number of even samples. – lance May 5 '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 5 '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 Chernick May 5 '12 at 14:12
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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.

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