Edgeworth expansions for compound Poisson processes and the bootstrap

  • Gutti Jogesh Babu
  • Kesar Singh
  • Yaning Yang
Edgeworth Expansion


One-term Edgeworth Expansions for the studentized version of compound Poisson processes are developed. For a suitably defined bootstrap in this context, the so called one-term Edgeworth correction by bootstrap is also established. The results are applicable for constructing second-order correct confidence intervals (which make correction for skewness) for the parameter “mean reward per unit time”.

Key words and phrases

Renewal reward processes Poisson process studentization confidence interval approximate cumulant non-lattice distribution one-term Edgeworth correction by bootstrap 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Babu, G. J. and Bose, A. (1988). Bootstrap confidence intervals,Statist. Probab. Lett.,7, 151–160.CrossRefMathSciNetGoogle Scholar
  2. Babu, G. J. and Singh, K. (1984). On one term Edgeworth correction by Efron’s bootstrap,Sankhyà Ser. A,46, 219–232.MathSciNetGoogle Scholar
  3. Babu, G. J. and Singh, K. (1985). Edgeworth expansions for sampling without replacement from finite populations,J. Multivariate Anal.,17, 261–278.CrossRefMathSciNetGoogle Scholar
  4. Bhattacharya, R. N. and Ghosh, J. K. (1978). On the validity of formal Edgeworth expansion,Ann. Statist.,6, 434–451.MathSciNetGoogle Scholar
  5. Bhattacharya, R. N. and Ranga Rao, R. (1986).Normal Approximation and Asymptotic Expansions, 2nd ed., Wiley, New York.MATHGoogle Scholar
  6. Bose, A. and Babu, G. J. (1991). Accuracy of the bootstrap approximation,Probab. Theory Related Fields,90, 301–316.CrossRefMathSciNetGoogle Scholar
  7. Hall, P. (1988). Theoretical comparison of bootstrap confidence intervals (with discussions),Ann. Statist.,16, 927–985.MathSciNetGoogle Scholar
  8. Hipp, C. (1985). Asymptotic expansions in the central limit theorem for compound and Markov processes,Z. Wahrsch. verw. Gebiete.,69, 361–385.CrossRefMathSciNetGoogle Scholar
  9. Kusuoka, S. and Yoshida, N. (2000). Malliavin calculus, geometric mixing, and expansions of diffusion functionals,Probab. Theory Related Fields,116, 457–484.CrossRefMathSciNetGoogle Scholar
  10. Kutoyants, Yu. A. (1998).Statistical Inference for Spatial Poisson Processes, Springer, New York.MATHGoogle Scholar
  11. Mykland, P. A. (1992). Asymptotic expansions and bootstrapping distributions for dependent variables: A martingale approach,Ann. Statist.,20, 623–654.MathSciNetGoogle Scholar
  12. Mykland, P. A. (1993). Asymptotic expansions for martingales,Ann. Probab.,21, 800–818.MathSciNetGoogle Scholar
  13. Mykland, P. A. (1995a). Martingale expansions and second order inference,Ann. Statist.,23, 707–731.MathSciNetGoogle Scholar
  14. Mykland, P. A. (1995b). Embedding and asymptotic expansions for martingales,Probab. Theory Related Fields,103, 475–492.CrossRefMathSciNetGoogle Scholar
  15. Singh, K. (1981). On asymptotic accuracy of Efron’s bootstrap,Ann. Statist.,9, 1187–1195.MathSciNetGoogle Scholar
  16. Yoshida, N. (2001). Malliavin calculus and martingale expansion,Bull. Sci. Math.,125, 431–456.CrossRefMathSciNetGoogle Scholar

Copyright information

© The Institute of Statistical Mathematics 2003

Authors and Affiliations

  • Gutti Jogesh Babu
    • 1
  • Kesar Singh
    • 2
  • Yaning Yang
    • 2
  1. 1.Department of StatisticsThe Pennsylvania State UniversityUniversity ParkUSA
  2. 2.Department of Statistics, Faculty of Arts and SciencesRutgers University, Hill Center Busch CampusPiscatawayUSA

Personalised recommendations