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I'm curious if there has been any work looking at order statistics for a distribution that is the combination of multiple distributions? For example, consider if X is a random such that with probability 1/2 it is drawn from an exponential distribution with λ = 1 and with probability 1/2 it is drawn from an exponential distribution with λ = 2. Is there a formula for the distribution of its order statistics for a random sample of X? Similarly, if I had certain order statistics, for example, the median, for two random samples one of which is drawn from an exponential distribution with λ = 1 and the other an exponential distribution with λ = 2, is there a way to approximate the median for a random sample of random variables that are drawn from either distribution with probability 1/2?

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You are asking about order statistics of mixture distributions. In your example we have

$$F_X(x) = \frac 12 F_1(x) + \frac 12 F_2(x)$$

and, say, the maximum order statistic from a sample of size $n$ from this random variable has distribution

$$F_{(n)}(x) = [F_X(x)]^n = \frac 1{2^n} [F_1(x) + F_2(x)]^n$$

This generalizes to convex combinations,

$$F_X(x) = \sum_{i=1}^kw_iF_i(x), \;\; \sum w_i =1 , \;w_i>0$$

$$F_{(n)}(x) = \left[\sum_{i=1}^kw_iF_i(x)\right]^n$$

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  • $\begingroup$ Thanks! If I wanted to estimate, say the 90th percentile of F(x), and I had the 90th percentile for two samples, one drawn from F1(x) and the other F2(X), could I estimate the 90th percentile of F(x) with them? $\endgroup$ – jeromefroe Feb 19 '17 at 19:53
  • $\begingroup$ The above is the distribution of the maximum order statistic. Look up the general expression for the distribution of an order statistic, see the comments in this answer, stats.stackexchange.com/a/262353/28746 and plug in the mixture distribution $\endgroup$ – Alecos Papadopoulos Feb 19 '17 at 19:58
  • $\begingroup$ Thanks @Alecos, so to use a sample quantile as an unbiased estimator of the true quantile of a distribution I need to show the expected value of the sample quantile is equal to that quantile of the distribution and to do that I can use the expression for the distribution of an order statistic? $\endgroup$ – jeromefroe Feb 19 '17 at 20:56
  • $\begingroup$ @jeromefroe Usually unbiasedness requires to use the density rather than the distribution function, which can be derived by differentiation, but whether you can obtain a closed form result for the integral is another matter. $\endgroup$ – Alecos Papadopoulos Feb 19 '17 at 21:02
  • $\begingroup$ yea, off the top of my head, I suspect a closed form expression wouldn't exist. Thanks for all the helpful feedback though! $\endgroup$ – jeromefroe Feb 19 '17 at 21:09

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