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International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 58738, 22 pages

The probability of large deviations for the sum functions of spacings

Ghulam Ishaq Khan Institute of Engineering Sciences and Technology, Swabi Topi-23460, N.W.F.P., Pakistan

Received 16 March 2004; Revised 23 November 2005; Accepted 28 November 2005

Copyright © 2006 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let 0=U0,nU1,nUn1,nUn,n=1 be an ordered sample from uniform [0,1] distribution, and Din=Ui,nUi1,n, i=1,2,,n; n=1,2,, be their spacings, and let f1n,,fnn be a set of measurable functions. In this paper, the probabilities of the moderate and Cramer-type large deviation theorems for statistics Rn(D)=f1n(nD1n)++fnn(nDnn) are proved. Application of these theorems for determination of the intermediate efficiencies of the tests based on Rn(D)-type statistic is presented here too.