Rate of convergence towards a Fréchet type limit distribution. (English) Zbl 0535.60022

Iglèsias Pereira, Helena

Ann. Sci. Univ. Clermont-Ferrand II 76, Probab. Appl. 1, 67-80 (1983).

Summary: Let \(\{X_ i\}\), \(i\in N\), be a sequence of i.i.d. random variables, and \(M_ n=\max(X_ 1,...,X_ n)\). It is well-known that \(F_{M_ n}(x)=F^ n\!_{X_ 1}(x)\), and that if there are attraction coefficients \(\{a_ n\}\), \(n\in N\), \((a_ n>0)\) and \(\{b_ n\}\), \(n\in N\), \((b_ n\in R)\) such that \(F^ n(a_ nx+b_ n)\to G(x)\), \(x\in C_ G\), then G is one of the three extreme value stable types: \(\Lambda(x)=\exp(-e^{-x})\) if \(-\infty<x<+\infty\) (Gumbel type), \(\Phi_{\alpha}(x)=\exp(-x^{-\alpha})\) if \(0<x<\infty\) (Fréchet type), \(\Psi_{\alpha}(x)=\exp(-(-x^{\alpha}))\) if \(-\infty<x<0\) (Weibull type). There are no definite results on the rate of convergence of \(F^ n\) towards the limiting form but in the case F is of normal type. Under mild conditions on the tailweight of F, we study the rate of convergence in the case of a Fréchet type limit distribution.

MSC:

60F05

Central limit and other weak theorems

Keywords:

extreme value stable types; rate of convergence

References:

[1]	Abramowitz , M. & Stegun , I.A. ( 1972 ) ” Handbook of Mathematical Functions ”. Dover , New York . · Zbl 0543.33001
[2]	Anderson , C.W. ( 1971 ) ” Contributions to the Asymptotic Theory of Extreme Values ”, Ph. D. Thesis, Imperial College , London .
[3]	Fisher , R.A. & Tippett , L.H. ( 1928 ) ” Limiting Forms of the Frequency Distribution of the Largest or Smallest Member of a Sample ”, Proceed. of the Cambridge Philosophical Society , vol. 24 . JFM 54.0560.05 · JFM 54.0560.05
[4]	Galambos , J. ( 1978 ) ” The Asymptotic Theory of Extreme Order Statistics ”, J. Wiley & Sons , New York . MR 489334 | Zbl 0381.62039 · Zbl 0381.62039
[5]	Gnedenko , B.V. ( 1943 ) ” Sur la Distribution Limite du Terme Maximum d’une Série Aléatoire ”, Ann. Math. , vol. 44 , 423 - 453 . MR 8655 | Zbl 0063.01643 · Zbl 0063.01643 · doi:10.2307/1968974
[6]	Gnedenko , B.V. & Kolmogorov , A.N. ( 1968 ) ” Limit Distributions for Sums of Independent Random Variables ”, Addison-Wesley , Cambridge (Mass.) . MR 233400 | Zbl 0056.36001 · Zbl 0056.36001
[7]	Gomes , I. ( 1982 ) ” Penultimate Limiting Forms in Extreme Value Theory ”, Centro de Estatistica e Aplicações da Univ. de Lisboa . · Zbl 0561.62015
[8]	Gomes . I. ( 1978 ) ” Problems in Extreme Value Theory ”, Ph. D. Thesis, Sheffield .
[9]	Iglésias . H. ( 1982 a) ” On the Domain of Attraction of Extreme Value Stable Distributions ”, a publicar.
[10]	Iglésias , H. ( 1982 b) ” Rate of Convergence Towards a Fréchet Type Limit Distribution ”, IX Jornadas Hispano-Lusas de Matemática, Salamanca . MR 798634 · Zbl 0535.60022
[11]	Iglésias , H. ( 1982 c) ” Great Age in Biometry ”, XI Conference International de Biometrie, Toulouse .