Abstract
We obtain weighted approximations by a Brownian bridge to permutation and exchangeable processes and to appropriately defined inverse processes. Our results provide as special cases useful weighted approximations to the uniform empirical and quantile processes and to generalized bootstrapped versions of these processes. A number of other applications are discussed. Our approach is based on the Skorokhod embedding for martingales.
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References
Bickel, P. J., and Freedman, D. A. (1981). Some asymptotic theory for the bootstrap.Ann. Stat. 9, 1196–1217.
Billingsley, P. (1968).Convergence of Probability Measures. Wiley, New York.
Csörgő, M., Csörgő, S., Horváth, L., and Mason, D. M. (1986). Weighted empirical and quantile processes.Ann. Probab. 14, 31–85.
Csörgő, S., and Mason, D. M. (1989). Bootstrapping empirical functions.Ann. Stat. 17, 1447–1471.
Efron, B. (1979). Bootstrap methods: Another look at the jackknife.Ann. Stat. 7, 1–26.
Freedman, D. A. (1982)Brownian Motion and Diffusion. Springer, New York.
Hájek, J., and Šidák, Z. (1967)Theory of Rank Tests, Academic Press, New York.
Hall, P., and Heyde, C. C. (1980).Martingale Limit Theory and Its Application. Academic Press, New York.
Koul, H. (1970). Some convergence theorems for ranks and weighted empirical cumulatives.Ann. Math. Stat. 41, 1768–1773.
Mason, D. M. (1991). A note on weighted approximations to the uniform empirical and quantile processes. In M. Hahn, D. Mason, and D. Weiner (Eds.),Sums, Trimmed Sums and Extremes. Birkhäuser, Boston, pp. 269–283.
Mason, D. M., and Newton, M. A. (1992). A rank statistics approach to the consistency of a general bootstrap.Ann. Stat., to appear.
Mason, D. M., and van Zwet, W. R. (1987). A refinement of the KMT inequality for the uniform empirical process.Ann. Probab. 15, 871–884.
Praestgaard, J. (1990). General bootstrap of empirical measures. Preprint.
Rubin, D. (1981). The Bayesian bootstrap.Ann. Stat. 9, 130–134.
Scott, D. J., and Huggins, R. M. (1983). On the embedding of processes in Brownian motion and the law of the iterated logarithm for reverse Martingales.Bull. Austr. Math. Soc. 27, 443–459.
Shorack, G. R. (1991). Embedding the sampling process at a rate.Ann. Probab. 19, 826–842.
Shorack, G. R., and Wellner, J. A. (1986).Empirical Processes with Applications to Statistics. Wiley, New York.
Wellner, J. A. (1978). Limit theorems for the ratio of the empirical distribution function to the true distribution function.Z. Wahrsch. verw. Gebiete 45, 73–88.
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Einmahl, U., Mason, D.M. Approximations to permutation and exchangeable processes. J Theor Probab 5, 101–126 (1992). https://doi.org/10.1007/BF01046780
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DOI: https://doi.org/10.1007/BF01046780