ASYMPTOTIC DISTRIBUTIONS OF LINEAR AND N
Résumé
The limit distributions of linear and non-linear combinations of the kn = o(n) order statistics of i.i.d. random variables whose
maximum belongs to the domain of attraction of the Gumbel law are obtained. Our results may be applied in actuarial studies,
estimation of scale-location parameters, estimation of squared deviation in tail of a distribution, robustness theory and
detection of the outliers in statistical data. It is also closely related to the moment estimator of Dekkers-Einmahl-de Hann
(1989) for the index of an extreme distribution. This study completes that of Necir (1990, 1991a, 1991b, 2000a, 2000b).
Références
Barnett, V. and B. Lewis (1994). Outliers in
Statistical Data.3 rd Ed. John Wiley & Sons.
[2] Beirlant, J. and J. L. Teugels (1987). Asymptotics of
Hill's estimator. Th. Probability Appl., 31, 463-469.
[3] Boos, D. (1979). A Differential for L-statistics. Ann.
Statist., 7, 955-959.
[4] Chernoff, H., J. Gastwirth and M. Johns (1967).
Asymptotic distributions of linear combinations of
functions of order statistics with applications to
estimation. Ann. Math. Statist.,38, 52-72.
[5] Csörgő, M. and P. Révész (1981). Strong
Approximations in Probability and Statistics.
Academic, New York.
[6] Csörgő M., S. Csörgő, L. Horváth and D. M. Mason
(1986). Weighted empirical and quantile processes.
Ann. Probab.,14; 31-85.
[7] Csörgő, S., P. Deheuvels and D. M. Mason (1985).
Kernel estimates of the tail index of distribution.
Ann. Statist.,14, 31-85.
[8] Csörgő S. and D. M. Mason (1989). Simple
estimators of the endpoint of a distribution. In
“Extreme Value Theory” (ed. J. Husler and R. D.
Reiss), Lecture Notes in Statistics, Spinger, New
York, 132-147.
[9] De Hann, L. (1970). On regular Variation and its
Application to the Weak Convergence of Sample
Extremes. Amsterdam: Math. Centre Tracts 32.
Asymptotic distributions of linear and non linear combinations of extreme order statistics
15
[10] Dekkers, A. L. M., Einmahl, J. H. J. and de Hann, L.
(1989). A moment estimator for the index of an
extreme-value distribution. Ann. Statist., 17, 1833-
1855.
[11] Deheuvels, P., E. Haeusler and D. M. Mason (1990).
On almost sure behavior of sums of extreme values
from a distribution in the domain of attraction of
Gumbel law. Bull. Sci. Math. Série No. 2, 114, 61-
95.
[12] Dixon W. J. and J.W. Tukey (1968). Approximate
behavior of the distribution of Winsorized t
(Trimming/Winsorisation 2). Technometrics, 10, 83-
98.
[13] Einmahl, J. H. J. and D. M. Mason (1988). Strong
limit theorems for weighted quantile process. Ann.
Probab., 16, 126-141.
[14] Falk, M. (1985). Asymptotic normality of kernel
quantile estimator. Ann. Statist., 13, 428-433.
[15] Falk, M. (1995). Some best parameter estimates for
distributions with finite endpoint. Statistics, 27, 115-
125.
[16] Fung, K. Y. and S. R. Paul (1985). Comparisons of
outlier detection procedures in Weibull or extreme
value distribution. Commun. Statist. Assn.,75, 395-
398.
[17] Galambos, J. (1987). The asymptotic Theory of
Extreme Order Statistics., 2nd edition, Malabar:
Kreiger.
[18] Gnedenko, B. (1943). Sur la distribution limité du
terme maximum d'une série aléatoire. Ann. Math. 44,
423-453.
[19] Hawkins, D. M. (1979). Fractiles of an extended
multiple outlier test. J. Statist. Comp. Sim., 8, 227-
236.
[20] Helmers, R. and F. Ruymgaart (1988). Asymptotic
normality of generalized L-statistics with unbounded
scores. J. Statist. Plann. Inference, 19, 43-53.
[21] Lo, G. S. (1989). A Note on the asymptotic
normality of sums of extreme value. J. Statist. Plann.
Inference, 22, 27-136, North-Holland.
[22] Mason, D. M. (1981). Asymptotic normality of
linear combinations of order statistics with a smooth
score function. Ann. Statist., 9, 899-904.
[23] Mason, D. M. and G. Shorack (1990). Necessary and
sufficient conditions for asymptotic normality of
trimmed L-statistics. J. Statist. Plann Inference, 25,
111-139.
[24] Necir, A. (1990). Asymptotic normality and law of
the iterated logarithm for linear combinations of
extreme values from a distribution with regularly
varying upper tail. Technical report, 123, L.S.T.A.,
Paris VI.
[25] Necir, A. (1991a). Sur le comportement
asymptotique des combinaisons linéaires de
statistiques d'ordre extrêmes issue d'une distribution
appartenant au domaine d'attraction de la loi de
Gumbel. C. R. Acd. Sci. Paris, t. 312, Série I, 159-
163.
[26] Necir, A. (1991b). Lois du logarithme itéré pour des
combinaisons linéaires de statistiques d'ordre
extrêmes des distributions appartenant au domaine
d'attraction de la loi de Gumbel. C. R. Acd. Sci.
Paris, t. 312, Série I, 245-250.
[27] Necir, A.(2000a). A Note on the functional law of
the iterated logarithm for non-linear combinations of
extreme order statistics. J. Nonpar Statist. (to be
published).
[28] Necir, A.(2000b). Asymptotic normality of weighted
L-statistics based upon extreme values.
CIMASI'2000, Casablanca, Morocco.
[29] Ruymgaart, F. and M. Van Zuijlen (1977).
Asymptotic normality of linear combinations of
order statistics in the non-i.i.d. case. Nedrl.Akad.
Wetensch. Proc. Ser. A 80, 5, 432-447.
[30] Seneta, E. (1975). Regularly varying functions.
Spinger-Verlag, Berlin.
[31] Serfling, R. J. (1980). Approximation Theorems of
Mathematical Statistics. Wiley, New York.}
[32] Shorack, G. and Wellner (1986). Empirical Process
with Applications to Statistics. John Wiley and Sons,
New York.
[33] Singh, K. (1981). On asymptotic representation and
approximation to normality of L-statistics. I.
Sankhya Ser. A, 43, 67-83.
[34] Stigler, S. (1969). Linear functions of order statistics.
Ann. Math. Statist., 40, 770-788.
[35] Stigler, S. (1974). Linear functions of order statistics
with smooth weight functions. Ann. Math. Statist., 2,
676-693.
[36] Tietjen, G. L. and R. H. Moore (1979). Some
Grubbs-type statistics for the detection of several
outliers. Technometrics, 14, 583-597.
[37] Teugels, J. L. (1984). Extreme values in insurance
mathematics, in Statistical Extremes and
Applications. (J. Tiago de Olivera, ed.) NATO ASI
Series. Reidel, Dordrecht, 253-259.
[38] Tobbal, K. (1995). A functional law of the iterated
logarithm for the Dekkers-Einmahl-De Hann tail
index estimator. J. Nonpar. Statist , 5, 145-156.
Statistical Data.3 rd Ed. John Wiley & Sons.
[2] Beirlant, J. and J. L. Teugels (1987). Asymptotics of
Hill's estimator. Th. Probability Appl., 31, 463-469.
[3] Boos, D. (1979). A Differential for L-statistics. Ann.
Statist., 7, 955-959.
[4] Chernoff, H., J. Gastwirth and M. Johns (1967).
Asymptotic distributions of linear combinations of
functions of order statistics with applications to
estimation. Ann. Math. Statist.,38, 52-72.
[5] Csörgő, M. and P. Révész (1981). Strong
Approximations in Probability and Statistics.
Academic, New York.
[6] Csörgő M., S. Csörgő, L. Horváth and D. M. Mason
(1986). Weighted empirical and quantile processes.
Ann. Probab.,14; 31-85.
[7] Csörgő, S., P. Deheuvels and D. M. Mason (1985).
Kernel estimates of the tail index of distribution.
Ann. Statist.,14, 31-85.
[8] Csörgő S. and D. M. Mason (1989). Simple
estimators of the endpoint of a distribution. In
“Extreme Value Theory” (ed. J. Husler and R. D.
Reiss), Lecture Notes in Statistics, Spinger, New
York, 132-147.
[9] De Hann, L. (1970). On regular Variation and its
Application to the Weak Convergence of Sample
Extremes. Amsterdam: Math. Centre Tracts 32.
Asymptotic distributions of linear and non linear combinations of extreme order statistics
15
[10] Dekkers, A. L. M., Einmahl, J. H. J. and de Hann, L.
(1989). A moment estimator for the index of an
extreme-value distribution. Ann. Statist., 17, 1833-
1855.
[11] Deheuvels, P., E. Haeusler and D. M. Mason (1990).
On almost sure behavior of sums of extreme values
from a distribution in the domain of attraction of
Gumbel law. Bull. Sci. Math. Série No. 2, 114, 61-
95.
[12] Dixon W. J. and J.W. Tukey (1968). Approximate
behavior of the distribution of Winsorized t
(Trimming/Winsorisation 2). Technometrics, 10, 83-
98.
[13] Einmahl, J. H. J. and D. M. Mason (1988). Strong
limit theorems for weighted quantile process. Ann.
Probab., 16, 126-141.
[14] Falk, M. (1985). Asymptotic normality of kernel
quantile estimator. Ann. Statist., 13, 428-433.
[15] Falk, M. (1995). Some best parameter estimates for
distributions with finite endpoint. Statistics, 27, 115-
125.
[16] Fung, K. Y. and S. R. Paul (1985). Comparisons of
outlier detection procedures in Weibull or extreme
value distribution. Commun. Statist. Assn.,75, 395-
398.
[17] Galambos, J. (1987). The asymptotic Theory of
Extreme Order Statistics., 2nd edition, Malabar:
Kreiger.
[18] Gnedenko, B. (1943). Sur la distribution limité du
terme maximum d'une série aléatoire. Ann. Math. 44,
423-453.
[19] Hawkins, D. M. (1979). Fractiles of an extended
multiple outlier test. J. Statist. Comp. Sim., 8, 227-
236.
[20] Helmers, R. and F. Ruymgaart (1988). Asymptotic
normality of generalized L-statistics with unbounded
scores. J. Statist. Plann. Inference, 19, 43-53.
[21] Lo, G. S. (1989). A Note on the asymptotic
normality of sums of extreme value. J. Statist. Plann.
Inference, 22, 27-136, North-Holland.
[22] Mason, D. M. (1981). Asymptotic normality of
linear combinations of order statistics with a smooth
score function. Ann. Statist., 9, 899-904.
[23] Mason, D. M. and G. Shorack (1990). Necessary and
sufficient conditions for asymptotic normality of
trimmed L-statistics. J. Statist. Plann Inference, 25,
111-139.
[24] Necir, A. (1990). Asymptotic normality and law of
the iterated logarithm for linear combinations of
extreme values from a distribution with regularly
varying upper tail. Technical report, 123, L.S.T.A.,
Paris VI.
[25] Necir, A. (1991a). Sur le comportement
asymptotique des combinaisons linéaires de
statistiques d'ordre extrêmes issue d'une distribution
appartenant au domaine d'attraction de la loi de
Gumbel. C. R. Acd. Sci. Paris, t. 312, Série I, 159-
163.
[26] Necir, A. (1991b). Lois du logarithme itéré pour des
combinaisons linéaires de statistiques d'ordre
extrêmes des distributions appartenant au domaine
d'attraction de la loi de Gumbel. C. R. Acd. Sci.
Paris, t. 312, Série I, 245-250.
[27] Necir, A.(2000a). A Note on the functional law of
the iterated logarithm for non-linear combinations of
extreme order statistics. J. Nonpar Statist. (to be
published).
[28] Necir, A.(2000b). Asymptotic normality of weighted
L-statistics based upon extreme values.
CIMASI'2000, Casablanca, Morocco.
[29] Ruymgaart, F. and M. Van Zuijlen (1977).
Asymptotic normality of linear combinations of
order statistics in the non-i.i.d. case. Nedrl.Akad.
Wetensch. Proc. Ser. A 80, 5, 432-447.
[30] Seneta, E. (1975). Regularly varying functions.
Spinger-Verlag, Berlin.
[31] Serfling, R. J. (1980). Approximation Theorems of
Mathematical Statistics. Wiley, New York.}
[32] Shorack, G. and Wellner (1986). Empirical Process
with Applications to Statistics. John Wiley and Sons,
New York.
[33] Singh, K. (1981). On asymptotic representation and
approximation to normality of L-statistics. I.
Sankhya Ser. A, 43, 67-83.
[34] Stigler, S. (1969). Linear functions of order statistics.
Ann. Math. Statist., 40, 770-788.
[35] Stigler, S. (1974). Linear functions of order statistics
with smooth weight functions. Ann. Math. Statist., 2,
676-693.
[36] Tietjen, G. L. and R. H. Moore (1979). Some
Grubbs-type statistics for the detection of several
outliers. Technometrics, 14, 583-597.
[37] Teugels, J. L. (1984). Extreme values in insurance
mathematics, in Statistical Extremes and
Applications. (J. Tiago de Olivera, ed.) NATO ASI
Series. Reidel, Dordrecht, 253-259.
[38] Tobbal, K. (1995). A functional law of the iterated
logarithm for the Dekkers-Einmahl-De Hann tail
index estimator. J. Nonpar. Statist , 5, 145-156.
Comment citer
NECIR, ABDELHAKIM; BRAHIMI, BRAHIM.
ASYMPTOTIC DISTRIBUTIONS OF LINEAR AND N.
Courrier du Savoir, [S.l.], v. 3, avr. 2014.
ISSN 1112-3338.
Disponible à l'adresse : >https://revues.univ-biskra.dz./index.php/cds/article/view/216>. Date de consultation : 14 nov. 2024
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