Sample quantiles and additive statistics: Information, sufficiency, estimation

The order of the increase in the Fisher information measure contained in a finite number k of additive statistics or sample quantiles, constructed from a sample of size n, as n → ∞, is investigated. It is shown that the Fisher information in additive statistics increases asymptotically in a manner linear with respect to n, if 2 + δ moments of additive statistics exist for some δ > 0. If this condition does not hold, the order of increase in this information is non-linear and the information may even decrease. The problem of asymptotic sufficiency of sample quantiles is investigated and some linear analogues of maximum likelihood equations are constructed. AMS classifications: Primary 62B10, 62E20, 62F10, 62F12.