×

zbMATH — the first resource for mathematics

Comparison between the rates of convergence of extremes under linear and under power normalization. (English) Zbl 1247.60030
Summary: By using the theory of the second-order regular variation, we study the rates of the weak convergence of the maximum order statistics under power normalization. The exact rates are obtained in the uniform metric and the total variation metric. The relationship between the rates of convergence under linear and under power normalization is derived. Some illustrative examples are given for comparing the rates of convergence.

MSC:
60F05 Central limit and other weak theorems
62E20 Asymptotic distribution theory in statistics
62E15 Exact distribution theory in statistics
62G30 Order statistics; empirical distribution functions
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Balkema AA, de Haan L (1990) A convergence rate in extreme-value theory. J Appl Probab 27: 577–585 · Zbl 0723.60056
[2] Billingsley P (1979) Probability and measure. Wiley, New York · Zbl 0411.60001
[3] Christoph G, Falk M (1996) A note on domains of attraction of p-max stable laws. Stat Probab Lett 28: 279–284 · Zbl 0855.60027
[4] Galambos J (1987) The asymptotic theory of extreme order statistics. Wiley, New York · Zbl 0634.62044
[5] de Haan L (1970) On regular variation and its applications to the weak convergence of sample, extremes mathematical center tract, vol 32. Mathematisch Centum, Amsterdam · Zbl 0226.60039
[6] de Haan L, Resnick S (1996) Second-order regular variation and rates of converges in extreme value theory. Ann Probab 24: 97–124 · Zbl 0862.60039
[7] Hall P (1979) On the rate of converges of normal extremes. J Appl Probab 16: 433–439 · Zbl 0403.60024
[8] Mohan NR, Ravi S (1992) Max domains of attraction of univariate and multivariate p-max stable laws. Theory Probab Appl 37: 632–643 · Zbl 0788.60033
[9] Nasri-Roudsari D (1999) Limit distributions of generalized order statistics under power normalization. Commun Stat Theory Methods 28(6): 1379–1389 · Zbl 1045.62508
[10] Pantcheva E (1985) Limit theorems for extreme order statistics under nonlinear normalization. Lecture Notes in Mathematics, vol 1155. Springer, Berlin, pp 284–309 · Zbl 0572.62023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.