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On the Edgeworth expansion for elementary polynomials based on trimmed samples

Published online by Cambridge University Press:  17 April 2009

Yuri V. Borovskikh  and
N. C. Weber
Yuri V. Borovskikh
Affiliation:
Department of Applied Mathematics, Transport University, Moskovsky Avenue, 9, 190031 St Petersburg, Russia
N. C. Weber
Affiliation:
School of Mathematics and Statistics F07, University of Sydney, New South Wales 2006, Australia
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This paper develops a one term Edgeworth expansion under minimal conditions for elementary symmetric polynomials of any degree based on trimmed samples. These statistics are special cases of trimmed U-statistics and natural extensions of the trimmed mean.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 71 , Issue 1 , February 2005 , pp. 155 - 166
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Bahadur, R.R., ‘A note on quantiles in large samples’, Ann. Math. Statist. 37 (1966), 577–580.CrossRefGoogle Scholar
[2]Bentkus, V., Götze, F., and van Zwet, W.R., ‘An Edgeworth expansion for symmetric statistics’, Ann. Statist. 25 (1997), 851–896.CrossRefGoogle Scholar
[3]Bickel, P.J., ‘On some robust estimates of location’, Ann. Math. Statist. 36 (1965), 847–858.CrossRefGoogle Scholar
[4]Borovskikh, Y.V., U-statistics in Banach spaces (VSP, Utrecht, The Netherlands, 1996).Google Scholar
[5]Borovskikh, Y.V., Sharp estimates of the rate of convergence for U-statistics, (Report 98–21) (School of Mathematics and Statistics, University of Sydney, 1998).Google Scholar
[6]Borovskikh, Y.V. and Weber, N.C., ‘Large deviation results for a U-statistical sum with product kernel’, Bull. Austral. Math. Soc. 63 (2001), 151–165.Google Scholar
[7]Dickson, L.E., Elementary theory of equations (Wiley, New York, 1914).Google Scholar
[8]van Es, A.J., ‘On the weak limits of elementary symmetric polynomials’, Ann. Probab. 14 (1986), 677–695.CrossRefGoogle Scholar
[9]van Es, A.J. and Helmers, R., ‘Elementary symmetric polynomials of increasing order’, Probab. Theory Related Fields 80 (1986), 21–35.CrossRefGoogle Scholar
[10]van Es, A.J., Helmers, R. and Huskova, M., ‘On a crossroad of resampling plans: bootstrapping elementary symmetric polynomials’, Statist. Neerlandica 54 (2000), 100–110.CrossRefGoogle Scholar
[11]Gribkova, N. and Helmers, R., ‘The empirical Edgeworth expansion for a studentized trimmed mean’, (CWI Report PNA-RO214, (http://db.cwi.nl/rapporten/), Probability Networks and Algorithms (2002), 1–21.Google Scholar
[12]Halasz, G. and Szekely, G.J., ‘On the elementary symmetric polynomials of independent random variables’, Acta Math. Acad. Sci. Hung. 28 (1976), 397–400.CrossRefGoogle Scholar
[13]Hall, P. and Padmanabhan, A.R., ‘On the bootstrap and the trimmed mean’, J. Multivariate Anal. 41 (1992), 132–153.Google Scholar
[14] Helmers, R., ‘Edgeworth expansions for trimmed linear combinations of order statistics’, in Proceedings of the Second Prague Symp. Asymptotic Statist., (Mandl, P. and Huskova, M., Editors) (North-Holland, Amsterdam, 1979), pp. 221–232.Google Scholar
[15]Helmers, R., ‘A Berry-Esseen theorem for linear combinations of order statistics’, Ann. Probab. 9 (1981), 342–347.Google Scholar
[16]Hoeffding, W., ‘Probability inequalities for sums of bounded random variables’, J. Amer. Statist. Assoc. 58 (1963), 13–30.Google Scholar
[17]Janssen, P., Serfling, R. and Veraverbeke, N., ‘Asymptotic normality of U-statistics based on trimmed samples’, J. Statist. Plann. Inference 16 (1987), 63–74.CrossRefGoogle Scholar
[18]Jing, B.Y. and Wang, Q., ‘Edgeworth expansion for U-statistics under minimal conditions’, Ann. Statist. 31 (2003), 1376–1391.CrossRefGoogle Scholar
[19]Putter, H. and van Zwet, W.R., ‘Empirical Edgeworth expansions for symmetric statistics’, Ann. Statist. 26 (1998), 1540–1569.Google Scholar
[20]Reiss, R.D., Approximate distributions of order statistics (Springer-Verlag, New York, 1989).CrossRefGoogle Scholar
[21]Szekely, G.J., ‘A limit theorem for elementary symmetric polynomials of independent random variables’, Z. Wahrsch. Verw. Gebiete 59 (1982), 355–359.Google Scholar
[22]de Wet, T., ‘Berry-Esseen results for the trimmed mean’, South African Statist. J. 10 (1976), 77–96.Google Scholar
[23]de Wet, T. and Venter, J.H., ‘An asymptotic representation of trimmed means with applications’, South African Statist. J. 8 (1974), 127–133.Google Scholar
[24] van Zwet, W.R., ‘Asymptotic expansions for the distribution functions of linear combinations of order statistics’, in Statistical decision theory and related topics II, (Moore, D.S. and Gupta, S.S., Editors) (Academic Press, New York, 1977), pp. 421–437.CrossRefGoogle Scholar
[25] van Zwet, W.R., ‘The Edgeworth expansion for linear combinations of uniform order statistics’, in Proceedings of the Second Prague Symp. Asymptotic Statist., (Mandle, P. and Huskova, M., Editors) (North-Holland, Amsterdam, 1979), pp. 93–101.Google Scholar