 Solution

ID: past-exam-of-the-statistics-course-of-the-university-of-oxford/2015/1/a/i/solution

r

and

n - r + 1

. We can derive the required moments directly rather than assuming the beta moment formulas. For an integer

j \geq 0

E [U_{(r)}^{j}] = \frac{n !}{( r - 1 )! ( n - r )!} \int_{0}^{1} u^{r + j - 1} (1 - u)^{n - r} d u = \frac{n ! ( r + j - 1 )!}{( r - 1 )! ( n + j )!} .

Here

B (a, b) = (a - 1)! (b - 1)! / (a + b - 1)!

a, b

; this identity also follows by repeated integration by parts. Taking

j = 1, 2

yields

E [U_{(r)}] = \frac{r}{n + 1}, E [U_{(r)}^{2}] = \frac{r ( r + 1 )}{( n + 1 ) ( n + 2 )} .

Subtract the square of the first moment from the second to obtain the variance:

E [U_{(r)}] = \frac{r}{n + 1}, Var (U_{(r)}) = \frac{r ( n - r + 1 )}{( n + 1 ) ^{2} ( n + 2 )} .

These are exact finite-sample identities for each uniform order statistic.

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