Hostname: page-component-6898cfcb55-th4rj Total loading time: 0 Render date: 2026-11-12T01:02:32.302Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic Expansions for Array Branching Processes with Applications to Bootstrapping

Part of:Markov processesStochastic processesDistribution theory

Published online by Cambridge University Press:  14 July 2016

T. N. Sriram
T. N. Sriram*
Affiliation:
University of Georgia
*
∗ Postal address: Department of Statistics, University of Georgia, Athens, GA 30602, USA.
Get accessRights & Permissions [Opens in a new window]

Abstract

Asymptotic expansions are obtained for the distribution function of a studentized estimator of the offspring mean sequence in an array branching process with immigration. The expansion result is shown to hold in a test function topology. As an application of this result, it is shown that the bootstrapping distribution of the estimator of the offspring mean in a sub-critical branching process with immigration also admits the same expansion (in probability). From these considerations, it is concluded that the bootstrapping distribution provides a better approximation asymptotically than the normal distribution.

Keywords

Edgeworth expansionsmartingalestest function topologybootstrappingarray branching processes

MSC classification

Primary:60J80: Branching processes (Galton-Watson, birth-and-death, etc.)62E20: Asymptotic distribution theory
Secondary:60G09: Exchangeability60G42: Martingales with discrete parameter

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 1 , March 1998, pp. 12 - 26
Copyright
Copyright © Applied Probability Trust 1998 

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.)

Article purchase

Temporarily unavailable

References

Billingsley, P. (1979). Probability and Measure. Wiley, New York.Google Scholar
Bose, A. (1988). Edgeworth correction by bootstrap in autoregressions. Ann. Statist. 16, 1709–1722.CrossRefGoogle Scholar
Chebyshev, P.L. (1890). Sur deux théorémes relatifs aux probabilités. Acta Math. 14, 305–315.Google Scholar
Chow, Y.S., and Teicher, H. (1978). Probability Theory: Independence, Interchangeability, Martingales. Springer, New York.CrossRefGoogle Scholar
Datta, S. (1992). A note on continuous Edgeworth expansions and the bootstrap. Sankhyā A 54, 171–182.Google Scholar
Datta, S., and Sriram, T.N. (1995). A modified bootstrap for branching processes with immigration. Stoch. Proc. Appl. 56, 275–294.CrossRefGoogle Scholar
Edgeworth, F.Y. (1883). The law of error. Phil. Mag. 16, 300–309.CrossRefGoogle Scholar
Edgeworth, F.Y. (1905). The law of error. Cambridge Phil. Trans. 20, 36–65, 113–141.Google Scholar
Goetze, F., and Hipp, C. (1983). Asymptotic expansions for sums of weakly dependent random vectors. Z. Wahrscheinlichkeitsth. 64, 211–239.CrossRefGoogle Scholar
Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.CrossRefGoogle Scholar
Hall, P., and Heyde, C.C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York.Google Scholar
Jensen, J.L. (1986). Asymptotic expansions for sums of dependent variables. Memoir 10. University of Aarhus.Google Scholar
Jensen, J.L. (1989). Asymptotic expansions for strongly mixing Harris recurrent Markov chains. Scand. J. Statist. 16, 47–64.Google Scholar
Mykland, P.A. (1992). Asymptotic expansions and bootstrapping-distributions for dependent variables: A martingale approach. Ann. Statist. 20, 623–654.CrossRefGoogle Scholar
Mykland, P.A. (1993). Asymptotic expansions for martingales. Ann. Prob. 2, 800–818.Google Scholar
Mykland, P.A. (1997). Martingale expansions and second order Inference. Ann Statist. 23, 707–731.Google Scholar
Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York.CrossRefGoogle Scholar
Singh, K. (1981). On the asymptotic accuracy of Efron's bootstrap. Ann. Statist. 9, 1187–1195.CrossRefGoogle Scholar
Sriram, T.N. (1994). Invalidity of bootstrap for critical branching processes with immigration. Ann. Statist. 22, 1013–1023.CrossRefGoogle Scholar
Sriram, T.N., Basawa, I.V., and Huggins, R.M. (1991). Sequential estimation for branching processes with immigration. Ann. Statist. 19, 2232–2243.CrossRefGoogle Scholar