Limit theorems for extremal residuals in a regression model with heavy tails of observation errors
Authors:
O. V. Ivanov and I. K. Matsak
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 88 (2014), 99-108
MSC (2010):
Primary 60G70, 62J05
DOI:
https://doi.org/10.1090/S0094-9000-2014-00921-7
Published electronically:
July 24, 2014
MathSciNet review:
3112637
Full-text PDF Free Access
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Additional Information
Abstract: Limit theorems for maximal residuals in a linear regression model with observation errors having heavy tails are obtained.
References
- O. V. Īvanov and Ī. K. Matsak, Limit theorems for extremal residuals in linear and nonlinear regression models, Teor. Ĭmovīr. Mat. Stat. 86 (2011), 69–80 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 86 (2013), 79–91. MR 2986451, DOI https://doi.org/10.1090/S0094-9000-2013-00890-4
- B. Gnedenko, Sur la distribution limite du terme maximum d’une série aléatoire, Ann. of Math. (2) 44 (1943), 423–453 (French). MR 8655, DOI https://doi.org/10.2307/1968974
- Janos Galambos, The asymptotic theory of extreme order statistics, John Wiley & Sons, New York-Chichester-Brisbane, 1978. Wiley Series in Probability and Mathematical Statistics. MR 489334
- M. R. Leadbetter, Georg Lindgren, and Holger Rootzén, Extremes and related properties of random sequences and processes, Springer Series in Statistics, Springer-Verlag, New York-Berlin, 1983. MR 691492
- V. V. Petrov, Summy nezavisimykh sluchaĭ nykh velichin, Izdat. “Nauka”, Moscow, 1972 (Russian). MR 0322927
- G. Peshkir and A. N. Shiryaev, Khinchin inequalities and a martingale extension of the sphere of their action, Uspekhi Mat. Nauk 50 (1995), no. 5(305), 3–62 (Russian); English transl., Russian Math. Surveys 50 (1995), no. 5, 849–904. MR 1365047, DOI https://doi.org/10.1070/RM1995v050n05ABEH002594
- William Feller, An introduction to probability theory and its applications. Vol. II., 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0270403
- James Pickands III, Moment convergence of sample extremes, Ann. Math. Statist. 39 (1968), 881–889. MR 224231, DOI https://doi.org/10.1214/aoms/1177698320
- Eugene Seneta, Regularly varying functions, Lecture Notes in Mathematics, Vol. 508, Springer-Verlag, Berlin-New York, 1976. MR 0453936
References
- O. V. Ivanov and I. K. Matsak, Limit theorems for extremal residuals in linear and nonlinear regression models, Teor. Imovir. Mat. Stat. 86 (2012), 69–80; English transl. in Theory Probab. Math. Statist. 86 (2013), 79–91. MR 2986451
- B. V. Gnedenko, Sur la distribution limit du terme maximum d‘une série aléatoire, Ann. Math. 44 (1943), 423–453. MR 0008655 (5:41b)
- J. Galambos, The Asymptotic Theory of Extreme Order Statistics, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York–Chichester–Brisbane, 1978. MR 489334 (80b:60040)
- M. R. Leadbetter, Georg Lindgren, and Holger Rootzen, Extremes and Related Properties of Random Sequences and Processes, Springer Series in Statistics, Springer-Verlag, New York, 1983. MR 691492 (84h:60050)
- V. V. Petrov, Sums of Independent Random Variables, “Nauka”, Moscow, 1972; English transl., Springer-Verlag, New York–Heidelberg, 1975. MR 0322927 (48:1288)
- G. Peshkir and A. N. Shiryaev, Khintchine inequalities and a martingale extension of the sphere of their action, Uspekhi Matematicheskikh Nauk 50 (1995), no. 3, 3–62; English transl. in Russian Mathematical Surveys 50 (1995), no. 5, 849–904. MR 1365047 (96k:60038)
- W. Feller, An Introduction to Probability Theory and its Applications, Vol. II, Second edition John Wiley & Sons Inc., New York–London–Sydney, 1971. MR 0270403 (42:5292)
- J. Pickands, Moment convergence of sample extremes, Ann. Math. Statist. 39 (1968), no. 3, 881–889. MR 0224231 (36:7275)
- Eugene Seneta, Regularly Varying Functions, Lecture Notes in Mathematics, Vol. 508, Springer-Verlag, Berlin, 1976. MR 0453936 (56:12189)
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Additional Information
O. V. Ivanov
Affiliation:
National Technical University of Ukraine “Kyiv Polytechnic Institute”, Department of Mathematical Analysis and Probability Theory, Peremogy Square, 37, Kyiv 03056, Ukraine
Email:
alexntuu@gmail.com
I. K. Matsak
Affiliation:
Kyiv National Taras Shevchenko University, Faculty for Cybernetics, Glushkov Avenue 2, Building 6, Kyiv 03127, Ukraine
Email:
mik@unicyb.kiev.ua
Keywords:
Regression model,
extremal values,
heavy tails
Received by editor(s):
July 17, 2012
Published electronically:
July 24, 2014
Article copyright:
© Copyright 2014
American Mathematical Society
