STA305H1 Lecture Notes - Lecture 14: Order Statistic, Triangular Matrix
18 Jan 2018
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Document Summary
Suppose that u1, , un are independent uniform random variables on the interval [0, 1] and de ne u(1) u(2) u(n) to be their order statistics. In class, we said that these order statistics could be represented as a function of n + 1 exponential random variables; speci cally, (u(1), u(2), , u(n)) d. E1 + + en+1! (1) where e1, , en+1 are independent exponential random variables with the same mean. (we will assume here that e1, , en+1 have mean 1. ) The representation given in (1) turns out to be very useful in class, we use it to derive the limiting distribution of central order statistics as well as obtaining the limiting distribution of spacings. The key to its utility is that it allows us to approximate order statistics and spacings by sums of independent random variables. The joint density of (u(1), u(2), , u(n)) is fu (y1, , yn) = n!
