Order Statistics
The sorted sample , with joint density on .
Definition 4.30 (Order Statistics)
Let be i.i.d. random variables with probability distribution function and density . Order them from the smallest to the largest as
Lete . Then, are called the order statistics of the random sample .
Minimum and maximum
For the minimum and maximum of the sample,
Joint density
For , each of the orderings of a realisation is equally likely, so
and differentiating,
Spacings of i.i.d. exponentials
Minima of independent exponentials are again exponential: if and are independent, then for ,
so ; more generally for independent .
Now let be i.i.d. with order statistics and spacings
The linear change of variables has Jacobian determinant , so
The joint density factorises, so the spacings are independent with .
Related
Stated in
- Definition 4.30 (Order Statistics)§4.7 Order Statistics of a Random Sample