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The asymptotic distribution of a cluster-index for i.i.d. normal random variables. (English) Zbl 1166.60030

Summary: In a sample variance decomposition, with components functions of the sample’s spacings, the largest component \(\tilde I_n\) is used in cluster detection. It is shown for normal samples that the asymptotic distribution of \(\tilde I_n\) is the Gumbel distribution.

MSC:

60G70 Extreme value theory; extremal stochastic processes
60F05 Central limit and other weak theorems
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[1] Chow, Y. S. and Teicher, H. (1988). Probability Theory : Independence , Interchangeability , Martingales , 2nd ed. Springer, New York. · Zbl 0652.60001
[2] David, H. A. and Nagaraja, H. N. (2003). Order Statistics , 3rd ed. Wiley, Hoboken, NJ. · Zbl 1053.62060
[3] Deheuvels, P. (1982). Strong limiting bounds for maximal uniform spacings. Ann. Probab. 10 1058-1065. · Zbl 0505.60033 · doi:10.1214/aop/1176993728
[4] Deheuvels, P. (1983). Upper bounds for k th maximal spacings. Z. Wahrsch. Verw. Gebiete 62 465-474. · Zbl 0488.60041 · doi:10.1007/BF00534198
[5] Deheuvels, P. (1984). Strong limit theorems for maximal spacings from a general univariate distribution. Ann. Probab. 12 1181-1193. · Zbl 0558.62018 · doi:10.1214/aop/1176993147
[6] Deheuvels, P. (1985). The limiting behaviour of the maximal spacing generated by an i.i.d. sequence of Gaussian random variables. J. Appl. Probab. 22 816-827. JSTOR: · Zbl 0584.60033 · doi:10.2307/3213949
[7] Devroye, L. (1981). Laws of the iterated logarithm for order statistics of uniform spacings. Ann. Probab. 9 860-867. · Zbl 0465.60038 · doi:10.1214/aop/1176994313
[8] Devroye, L. (1982). A log log law for maximal uniform spacings. Ann. Probab. 10 863-868. · Zbl 0491.60030 · doi:10.1214/aop/1176993799
[9] Devroye, L. (1984). The largest exponential spacing. Utilitas Math. 25 303-313. · Zbl 0547.60034
[10] Diaconis, P. and Freedman, D. (1984). Asymptotics of graphical projection pursuit. Ann. Statist. 12 793-815. · Zbl 0559.62002 · doi:10.1214/aos/1176346703
[11] Ibragimov, I. A. and Linnik, Y. V. (1971). Independent and Stationary Sequences of Random Variables . Wolters-Noordhoff Publishing, Groningen. With a supplementary chapter by I. A. Ibragimov and V. V. Petrov, Translation from the Russian edited by J. F. C. Kingman. · Zbl 0219.60027
[12] Nariaki, S. and Akihide, G. (1985). Pearson diagrams for truncated normal and truncated Weibull distributions. Biometrika 72 219-222.
[13] Pyke, R. (1965). Spacings. (With discussion.) J. Roy. Statist. Soc. Ser. B 27 395-449. JSTOR: · Zbl 0144.41704
[14] Slud, E. (1977/78). Entropy and maximal spacings for random partitions. Z. Wahrsch. Verw. Gebiete 41 341-352. · Zbl 0353.60019 · doi:10.1007/BF00533604
[15] Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics . Wiley, New York. · Zbl 0538.62002
[16] Yatracos, Y. G. (1998). Variance and clustering. Proc. Amer. Math. Soc. 126 1177-1179. JSTOR: · Zbl 0896.62062 · doi:10.1090/S0002-9939-98-04524-9
[17] Yatracos, Y. G. (2007). Cluster identification via projection pursuit. Unpublished manuscript.
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