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Say i have $X_{ij} \sim Exp(\theta)$ for $i=1...n,\;\;j=1...k$. Afterwards I derive $k$ medians for $k$ groups of size $n$ (even number). The median is a random variable say $M_j$. Do we have any idea about what distribution does this random variable have (theoretically) and furthermore how to compute its c.d.f?

If we do not, in order to approach the unknown distribution, do I have a chance by simply simulating the procedure above?

Gerasimos Chasapis
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  • https://en.wikipedia.org/wiki/Order_statistic#The_joint_distribution_of_the_order_statistics_of_an_absolutely_continuous_distribution – Sextus Empiricus Mar 06 '20 at 07:13
  • The density of the median can be derived from the cdf and density of the sample. It just happens that the Gamma cdf is not available in closed form, even though it can be approximated through implementations of the incomplete Gamma function. – Xi'an Mar 06 '20 at 13:24
  • @Xi'an question updated. May I have a mathematical explanation please, with refrences if possible? It would be highly appreciated! – Gerasimos Chasapis Mar 06 '20 at 16:06
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    You can use some R functions from this posts: https://stats.stackexchange.com/questions/441194/how-to-graph-distribution-of-order-statistics/441232#441232, https://stats.stackexchange.com/questions/303425/variance-of-sample-median-of-normal-distribution/306429#306429 – kjetil b halvorsen Mar 06 '20 at 16:20
  • Also, median is an order statistic, so search this site. – kjetil b halvorsen Mar 06 '20 at 16:27
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    The reference is the Wikipedia page! – Xi'an Mar 06 '20 at 17:26

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