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Suppose you just have some sample medians, and want to estimate the population median. Assume the samples are random, independent of each other, and of the same size.

Should you use the mean or the median of the sample medians? Does it give an unbiased estimator?

(Asking for a friend.)

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    $\begingroup$ Although this may be a duplicate of stats.stackexchange.com/q/4462, I don't like the answer there. It won't work correctly in many cases. The essence of the problem is revealed by considering extreme cases: they show how difficult it can be to combine medians into some overall estimator. But if you could be more specific about your situation, then we could focus on answers that might work specifically for you. Are these random samples? Are they independent of each other? Are they all the same or different sizes? What assumptions can you make about the population distribution? $\endgroup$
    – whuber
    Commented Feb 22, 2018 at 0:55
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    $\begingroup$ Assume the samples are random, independent, and of the same size. $\endgroup$ Commented Feb 22, 2018 at 3:10
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    $\begingroup$ Why do you need to decide in advance? Your sample medians are new data. You can take their mean and median and see if they agree closely or not. You can look at (including plot) the distribution of medians and judge what would be a good summary. Although the question is posed in abstraction there may be extra knowledge from context that you could bring to bear. $\endgroup$
    – Nick Cox
    Commented Feb 22, 2018 at 17:08

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  1. Of those two choices, use the mean of the sample medians. These are samples from the distribution of the sample median, which is a roughly normal distribution. So in most cases, the mean of sample medians will be a more efficient estimator than the median of sample medians.

  2. This mean will usually be a biased estimator of the original population median, at least when that original population is asymmetric. Eg suppose the original population has an exponential distribution with mean 1, and the sample medians come from samples of size 3. Then the sample medians have PDF $6e^{-2x}(1-e^{-x})$, so they have mean $5/6$ even though the actual population median is $\ln 2$.

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