Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-28T20:19:37.976Z Has data issue: false hasContentIssue false
  • English
  • Français

Weakly approaching sequences of random distributions

Part of: Distribution theory Probability theory on algebraic and topological structures Stochastic processes Parametric inference Nonparametric inference

Published online by Cambridge University Press:  14 July 2016

Yuri Belyaev  and
Sara Sjöstedt-de Luna
Yuri Belyaev*
Affiliation:
Umeå University
Sara Sjöstedt-de Luna*
Affiliation:
Umeå University
*
∗Postal address: Department of Mathematical Statistics, Umeå University, S-901 87 Umeå, Sweden
∗Postal address: Department of Mathematical Statistics, Umeå University, S-901 87 Umeå, Sweden
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce the notion of weakly approaching sequences of distributions, which is a generalization of the well-known concept of weak convergence of distributions. The main difference is that the suggested notion does not demand the existence of a limit distribution. A similar definition for conditional (random) distributions is presented. Several properties of weakly approaching sequences are given. The tightness of some of them is essential. The Cramér-Lévy continuity theorem for weak convergence is generalized to weakly approaching sequences of (random) distributions. It has several applications in statistics and probability. A few examples of applications to resampling are given.

Keywords

Weak convergenceweakly approaching sequencesresamplingbootstrapcontinuity theoremLévy metricuniform metric

MSC classification

Primary: 60B10: Convergence of probability measures 60G57: Random measures 62E20: Asymptotic distribution theory 62F12: Asymptotic properties of estimators 62G09: Resampling methods
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 3 , September 2000 , pp. 807 - 822
Copyright
Copyright © Applied Probability Trust 2000 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research supported by the Bank of Sweden Tercentenary Foundation, and by MISTRA, the Foundation of Strategic Environmental Research

References

Athreya, K. B. (1987). Bootstrap, of the mean in the infinite variance case. Ann. Statist. 15, 724–731.CrossRefGoogle Scholar
Belyaev, Y. K. (1995). Bootstrap, Resampling and Mallows Metric. Lecture Notes 1, Dept Mathematical Statistics, Umeì University, Sweden.Google Scholar
Belyaev, Y. K., and Nilsson, L. (1997). Parametric maximum likelihood estimators and resampling. Res. Rept 1997-15, Dept Mathematical Statistics, Umeì University, Sweden.Google Scholar
Belyaev, Y. K., and Seleznjev, O. (2000). Approaching in distribution with applications to resampling of stochastic processes. Scand. J. Statist. 27, 371–384.Google Scholar
Belyaev, Y. K. and Sjöstedt, S. (1997). Resampling from independent heterogeneous random variables with varying mean values. Theory Stoch. Proc. 3, 121–131.Google Scholar
Bickel, P. J., and Freedman, D. A. (1981). Some asymptotic theory for the bootstrap. Ann. Statist. 9, 1196–1217.Google Scholar
Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Cramér, H. (1946). Mathematical Methods of Statistics. Princeton University Press.Google Scholar
Davison, A. C., and Hinkley, D. V. (1997). Bootstrap Methods and their Application. Cambridge University Press.Google Scholar
Dudley, R. M. (1989). Real Analysis and Probability. Chapman and Hall, New York.Google Scholar
Efron, B., and Tibshirani, R. J. (1993). An Introduction to the Bootstrap. Chapman and Hall, New York.Google Scholar
Klebanov, L. B., Rachev, S. T., and Szekely, G. J. (1999). Pre-limit theorems and their applications. Acta Appl. Math. 58, 159–174.Google Scholar
Nilsson, L., and Belyaev, Y. K. (1998). Application of resampling to exponential and logistic regression. Res. Rept 1998-1, Dept Mathematical Statistics, Umeì University, Sweden.Google Scholar
Parthasarathy, K. R. (1977). Introduction to Probability and Measure. Macmillan, Delhi.Google Scholar
Rachev, S. T. (1991). Probability Metrics and the Stability of Stochastic Models. John Wiley, Chichester.Google Scholar
Searly, S. R. (1987). Linear Models for Unbalanced Data. John Wiley, New York.Google Scholar
Shao, J., and Tu, D. (1995). The Jackknife and Bootstrap. Springer, New York.Google Scholar
Shiryayev, A. N. (1996). Probability, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Sjöstedt, S. (2000). Resampling m-dependent random variables with applications to forecasting. Scand. J. Statist. 27, 543–561.CrossRefGoogle Scholar
Van der Vaart, A. W., and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York.Google Scholar
Wu, C. F. J. (1986). Jackknife, bootstrap and other resampling methods in regression analysis. Ann. Statist. 14, 1261–1295.Google Scholar