A Berry-Esseen Bound for Linear Combinations of Order Statistics

Abstract

The rate of convergence of the distribution function of a linear combination of order statistics of n independent and identically distributed random variables with a common distribution function F to its normal limit is investigated. Under the assumptions

some α ₁, , α ₂, β ₂ > 0 and

with some 0 ≤ κ < 4/3 and appropriate moment conditions a Berry-Esseen bound is given. If the coefficients are generated by a sequence of weight functions of a special structure, then the rate is shown to be . Finally, the result is applied for a statistic, which is widely used in auditing.

Keywords:

• Linear combinations of order statistics
• Berry-Esseen bound
• Stringer bound

Research supported by the Limperg Institute, which is the Interuniversity Research Institute for Accountancy in the Netherlands.

Research supported by the Limperg Institute, which is the Interuniversity Research Institute for Accountancy in the Netherlands.

Notes

Research supported by the Limperg Institute, which is the Interuniversity Research Institute for Accountancy in the Netherlands.