Abstract
It is proved that under fairly general von Mises-type conditions on the underlying distribution, the intermediate order statistics, properly standardized, converge uniformly over all Borel sets to the standard normal distribution. This closes the gap between central order statistics and extremes, where uniform convergence under mild conditions is well-known.
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References
Balkema, A. A. and de, Haan, L. (1972). On R. von Mises' condition for the domain of attraction of exp (−e -x), Ann. Math. Statist., 43, 1352–1354.
Balkema, A. A. and de, Haan, L. (1978a). Limit distributions for order statistics I, Theory Probab. Appl. 23, 77–92.
Balkema, A. A. and de, Haan, L. (1978b). Limit distributions for order statistics II, Theory Probab. Appl., 23, 341–358.
Chibisov, D. M. (1964). On limit distributions for order statistics, Theory Probab. Appl., 9, 142–148.
Cooil, B. (1985). Limiting multivariate distributions of intermediate order statistics, Ann. Probab., 13, 469–477.
de Haan, L. (1975). On Regular Variation and Its Application to the Weak Convergence of Sample Extremes, 3rd ed., Mathematical Centre Tracts, Vol. 32, Amsterdam.
de, Haan, L. and Resnick, S. I. (1982). Local limit theorems for sample extremes, Ann. Probab., 10, 396–413.
Falk, M. (1985). Uniform Convergence of Extreme Order Statistics, Habilitationsschrift, University of Siegen.
Falk, M. (1986). Rates of uniform convergence of extreme order statistics, Ann. Inst. Statist. Math., 38, 245–262.
Galambos, J. (1987). The Asymptotic Theory of Extreme Order Statistics, 2nd ed., Krieger, Melbourne, Florida.
Gnedenko, B. (1943). Sur la distribution limite du terme maximum d'une série aléatoire, Ann. Math., 44, 423–453.
Hall, P. (1979). On the rate of convergence of normal extremes, J. Appl. Probab., 16, 433–439.
Ikeda, S. and Matsunawa, T. (1972). On the uniform asymptotic joint normality of sample quantiles, Ann. Inst. Statist. Math., 24, 33–52.
Ikeda, S. and Matsunawa, T. (1976). Uniform asymptotic distribution of extremes, Essays in Probability and Statistics, (eds. S., Ikeda et al.), 419–432, Shinko Tsusho, Tokyo.
Pickands, J.III (1967). Sample sequences of maxima, Ann. Math. Statist., 38, 1570–1574.
Reiss, R.-D. (1976). Asymptotic expansions for sample quantiles, Ann. Probab., 4, 249–258.
Reiss, R.-D. (1981). Uniform approximation to distributions of extreme order statistics, Adv. in Appl. Probab., 13, 533–547.
Smirnov, N. V. (1952). Limit distributions for the terms of a variational series, Amer. Math. Soc. Transl., 67, 82–143.
Smirnov, N. V. (1967). Some remarks on limit laws for order statistics, Theory Probab. Appl., 12, 337–339.
Sweeting, T. J. (1985). On domains of uniform local attraction in extreme value theory, Ann. Probab., 13, 196–205.
von Mises, R. (1936). La distribution de la plus grande de n valeurs, reprinted in Selected Papers II, Amer. Math. Soc., Providence, Rhode Island, 1954, 271–294.
Weiss, L. (1969). The asymptotic joint distribution of an increasing number of sample quantiles, Ann. Inst. Statist. Math., 21, 257–263.
Weiss, L. (1971). Asymptotic inference about a density function at an end of its range, Naval Res. Logist. Quart., 18, 111–114.
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Falk, M. A note on uniform asymptotic normality of intermediate order statistics. Ann Inst Stat Math 41, 19–29 (1989). https://doi.org/10.1007/BF00049107
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DOI: https://doi.org/10.1007/BF00049107