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Weak and strong law results for a function of the spacings

  • William P. McCormick (a1)

Abstract

Let be i.i.d. uniform on (0,1) random variables and define Si,n = Ui ,n–1 Ui– 1,n–1, i = 1, · ··, n where the Ui –n–1 are the order statistics from a sample of size n – 1 and U 0,n–1 =0 and Un,n– 1 = 1. The Si,n are called the spacings divided by U 1,· ··,Un– 1. For a fixed integer l, set . Exact and weak limit results are obtained for the Ml,n. Further we show that with probability 1 This extends results of Cheng.

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Postal address: Department of Statistics and Computer Science, Graduate Studies Building, The University of Georgia, Athens, GA 30602, USA.

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Research supported by the National Science Foundation under Grant MCS8202259 and by AFOSR Grant No. F49620 82 C 0009.

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References

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[1] Balkema, A. A. and De Haan, L. (1978) Limit distributions for order statistics. I. Theory Prob. Appl. 23, 7792.
[2] Balkema, A. A. and De Haan, L. (1978) Limit distributions for order statistics. II. Theory Prob. Appl. 23, 341358.
[3] Bender, E. A. (1974) Asymptotic methods in enumeration. SIAM Review 16, 485515.
[4] Canfield, E. R. and Mccormick, W. P. (1983) Exact and limiting distributions for sustained maxima. J. Appl. Prob. 20, 803813.
[5] Cheng, S. (1983) On a problem concerning spacings. Center for Stochastic Processes Tech. Rept. # 27, Univ. of North Carolina, Chapel Hill, N.C.
[6] Chow, Y. S., Geman, S. and Wu, L. D. (1983) Consistent cross-validated density estimation. Ann. Statist. 11, 2538.
[7] Leadbetter, M. R., Lindgren, G. and Rootzen, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York.
[8] Marron, J. S. (1983) An asymptotic efficient solution to the bandwidth problem of kernel density estimation. Inst. Stat. Mimeo Series # 1518, Univ. of North Carolina, Chapel Hill, N.C.
[9] Pyke, R. (1965) Spacings. J. R. Statist. Soc. B 27, 395436.

Keywords

Weak and strong law results for a function of the spacings

  • William P. McCormick (a1)

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