Uniform approximation to distributions of extreme order statistics
- 1 September 1981
- journal article
- Published by Cambridge University Press (CUP) in Advances in Applied Probability
- Vol. 13 (3) , 533-547
- https://doi.org/10.2307/1426784
Abstract
This paper deals with asymptotic expansions of the distribution of the kth-largest order statistic Zn–k+1:n for the sample size n. These expansions establish higher-order approximations which hold uniformly over all Borel sets. In the particular case of the distribution of Zn–k+1:n under the uniform distribution and the exponential distribution, the approximating measures are linear combinations of ‘negative’ gamma distributions and linear combinations of extreme-value distributions. These results can be extended to the case of the joint distribution of the k largest order statistics. A numerical comparison to a different asymptotic expansion is given where the normal distribution is the leading term.Keywords
This publication has 6 references indexed in Scilit:
- The rate of convergence in law of the maximum of an exponential sampleStatistica Neerlandica, 1979
- On the rate of convergence of normal extremesJournal of Applied Probability, 1979
- Extreme value theory in applied probabilityAdvances in Applied Probability, 1979
- Asymptotic Expansions for Sample QuantilesThe Annals of Probability, 1976
- Asymptotic inference about a density function at an end of its rangeNaval Research Logistics Quarterly, 1971
- APPROXIMATE FORMULAE FOR THE STATISTICAL DISTRIBUTIONS OF EXTREME VALUESBiometrika, 1958