Fisher information in order statistics

When we have n independently and identically distributed observations, it is an interesting question how the Fisher information is distributed among order statistics. The recipe for the Fisher information in order statistics is easy, but the detailed calculation has been known to be complicated. An indirect approach, using a decomposition of the Fisher information in order statistics, simplifies the calculation. Some recurrence relations for the Fisher information in order statistics are derived that facilitate the calculation. The Fisher information in the first r order statistics is an r multiple integral, but it can be simplified to just a double integral by using the decomposition. A recurrence relation further simplifies the double integral to a sum of single integrals. An “information plot” is suggested, from which we can read at once the Fisher information in any set of consecutive order statistics for a parametric distribution.