• Bernoulli
  • Volume 6, Number 5 (2000), 809-834.

Bootstrap relative errors and sub-exponential distributions

Andrew T.A. Wood

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For the purposes of this paper, a distribution is sub-exponential if it has finite variance but its moment generating function is infinite on at least one side of the origin. The principal aim here is to study the relative error properties of the bootstrap approximation to the true distribution function of the sample mean in the important sub-exponential cases. Our results provide a fairly general description of how the bootstrap approximation breaks down in the tail when the underlying distribution is sub-exponential and satisfies some very mild additional conditions. The broad conclusion we draw is that the accuracy of the bootstrap approximation in the tail depends, in a rather sensitive way, on the precise tail behaviour of the underlying distribution. Our results are applied to several sub-exponential distributions, including the lognormal. The lognormal case is of particular interest because, as the simulation studies of Lee and Young have shown, bootstrap confidence intervals can have very poor coverage accuracy when applied to data from the lognormal.

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Bernoulli, Volume 6, Number 5 (2000), 809-834.

First available in Project Euclid: 6 April 2004

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Edgeworth expansion moderate deviations percentile method tail probability


Wood, Andrew T.A. Bootstrap relative errors and sub-exponential distributions. Bernoulli 6 (2000), no. 5, 809--834.

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  • [1] Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987) Regular Variation. Cambridge: Cambridge University Press.
  • [2] Booth, J.G., Hall, P. and Wood, A.T.A. (1994) On the validity of Edgeworth and saddlepoint approximations. J. Multivariate Anal., 51, 121-138.
  • [3] Efron, B. (1979) Bootstrap methods: another look at the jacknife. Ann. Statist., 7, 1-26.
  • [4] Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. New York: Chapman & Hall.
  • [5] Hahn, M.G. and Klass, M.J. (1997) Approximation of partial sums of arbitrary I.I.D. random variables and the precision of the usual exponential upper bound. Ann. Probab., 25, 1451-1470.
  • [6] Hall, P. (1986) On the bootstrap and confidence intervals. Ann. Statist., 14, 1431-1452.
  • [7] Hall, P. (1988) Theoretical comparison of bootstrap confidence intervals (with discussion). Ann. Statist., 16, 927-985.
  • [8] Hall, P. (1990) On the relative performance of bootstrap and Edgeworth approximations of a distribution function. J. Multivariate Anal., 35, 108-129.
  • [9] Hall, P. (1992) The Bootstrap and Edgeworth Expansion. New York: Springer-Verlag.
  • [10] Jing, B.Y. (1992) Saddlepoint and Edgeworth approximations and their applications to resampling. Unpublished Ph.D. thesis, University of Sydney.
  • [11] Jing, B.Y., Feuerverger, A. and Robinson, J. (1994) On the bootstrap saddlepoint approximation. Biometrika, 81, 211-215.
  • [12] Lee, S.M.S. and Young, G.A. (1995) Asymptotic iterated bootstrap confidence intervals. Ann. Statist., 23, 1301-1330.
  • [13] Nagaev, A.V. (1969) Limit theorems for large deviations when Cramér's conditions are violated (in Russian). Izv. Akad. Nauk. UzSSR Ser. Fiz.-Mat. Nauk., 6, 17-22.
  • [14] Nagaev, S.V. (1973) Large deviations for sums of independent random variables. In J. Kozesnik (ed.), Transactions of the Sixth Prague Conference on Information Theory, Statistical Decision Functions, Random Processes, pp. 657-674, Prague: Academia.
  • [15] Nagaev, S.V. (1979) Large deviations of sums of independent random variables. Ann. Probab., 7, 745- 789.
  • [16] Petrov, V.V. (1975) Sums of Independent Random Variables. Berlin: Springer-Verlag.
  • [17] Petrov, V.V. (1995) Limit Theorems of Probability Theory, Oxford Studies in Probability 4. Oxford: Clarendon Press.
  • [18] Pitman, E.J.G. (1980) Subexponential distribution functions. J. Austral. Math. Soc. Ser. A, 29, 337- 347.
  • [19] Singh, K. (1981) On the asymptotic accuracy of Efron's bootstrap. Ann. Statist., 9, 1189-1195.
  • [20] Teugels, J.L. (1975) The class of subexponential distributions. Ann. Probab., 3, 1000-1011.