Abstract
Let X1,X2,... be a sequence of independent copies (s.i.c) of a real random variable (r.v.) X ⩾ 1, with distribution function df F(x) = ℙ(X ⩽ x) and let X1,n ⩽ X2,n ⩽ ⋯ ⩽ Xn,n be the order statistics based on the n ⩾ 1 first of these observations. The following continuous generalized Hill process
τ < 0, 1 ⩽ k ⩽ n, has been introduced as a continuous family of estimators of the extreme value index, and largely studied for statistical purposes with asymptotic normality results restricted to τ < 1/2. We extend those results to 0 > τ ⩽ 1/2 and show that asymptotic normality is still valid for τ = 1/2. For 0 > τ < 1/2, we get non Gaussian asymptotic laws which are closely related to the Riemann function \(\xi (s) = \sum\nolimits_{n = 1}^\infty {{n^{ - s}},s > 1}\).
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References
Csörgő, M., Csörgő, S., Horvàth, L. and Mason, M. (1986). Weighted empirical and quantile processes. Ann. Probab. 14, 31–85.
de Haan, L. and Ferreira A. (2006). Extreme value theory: An introduction. Springer.
de Haan, L. (1970). On regular variation and its application to the weak convergence of sample extremes. Mathematical Centre Tracts, 32, Amsterdam.
De Haan, L. and Resnick, S.I. (1980). A simple asymptotic estimate for the index of a stable law. J. Roy. Statist. Soc., B, 83–87.
Diop Aliou and Lo G.S. (1990). Generalized Hill’s Estimator. Far East J. Theor. Statist., 20 (2), pp. 129–149.
Diop, A. and G.S. Lo. (2009). Ratio of Generalized Hill’s Estimator and its asymptotic normality theory. Math. Method. Statist., 18 (2), pp. 117–133.
Galambos, J. (1985). The Asymptotic theory of Extreme Order Statistics. Wiley, New York.
Hill, B.M. (1975). A simple general approach to the inference about the tail index of a distribution. Ann. Statist. 3, 1163–1174.
Lo, G.S. (1992). Sur la caractérisation empirique des extrêmes. C. R. Math. Rep. Acad. Sci. Canada, XIV, no 2, 3, 89–94.
Lo, G.S. (1986). Sur quelques estimateurs de l’index d’une loi de Pareto : Estimation de Deheuvels-Csörgő-Mason, de De Haan-Resnick et lois limites pour des sommes de valeurs extrêmes pour une variable dans le domaine de Gumbel. Thèse de Doctotat. Université Paris-6. France.
Pickands III, J. (1975). Statistical inference using extreme order statistics. Ann. Statist., 3, 119–131.
Resnick, S.I. (1987). Extreme Values, Regular Variation and Point Processes. Springer-Verbag, New York.
Shorack G.R. and Wellner J.A. (1986). Empirical Processes with Applications to Statistics. Wiley-Interscience, New York.
Valiron, G. (1990). Cours d’Analyse Mathématique : Théorie des fonctions. 3ième Edition. Masson, Paris.
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Lo, G.S., Deme, E.H. & Diop, A. On the Generalized Hill Process for Small Parameters and Applications. J Stat Theory Appl 12, 21–38 (2013). https://doi.org/10.2991/jsta.2013.12.1.3
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DOI: https://doi.org/10.2991/jsta.2013.12.1.3
Keywords
- Extreme values theory
- Asymptotic distribution
- Functional Gaussian and nongaussian laws
- Uniform entropy numbers
- Asymptotic tightness
- Stochastic process of estimators of extremal index
- Sowly and regularly varying functions