Edgeworth - Entwicklungen fuer Lineare Rangstatistiken (Edgeworth Expansions for Linear Rank Statistics)

摘要

The conditions which a matrix has to fulfil in order that its distribution function of the standardized linear rank statistics has Edgeworth expansions of first and second order with asymptotically suffiently small residual elements were investigated. The applied method is a combination of the Stein method and an extension of the Bolthausen method. The determined conditions are very similar to the necessary and sufficient conditions for the distribution function of a sum of independent and identically distributed random variables to have Edgeworth expansions. However, these conditions are often difficult to prove directly. They can be verified for simple linear rank statistics, and are shown to be much more general than the presently known conditions.