The Annals of Statistics

A Kiefer–Wolfowitz comparison theorem for Wicksell’s problem

Xiao Wang and Michael Woodroofe

Full-text: Open access

Abstract

We extend the isotonic analysis for Wicksell’s problem to estimate a regression function, which is motivated by the problem of estimating dark matter distribution in astronomy. The main result is a version of the Kiefer–Wolfowitz theorem comparing the empirical distribution to its least concave majorant, but with a convergence rate $n^{-1}\log n$ faster than $n^{-2/3}\log n$. The main result is useful in obtaining asymptotic distributions for estimators, such as isotonic and smooth estimators.

Article information

Source
Ann. Statist., Volume 35, Number 4 (2007), 1559-1575.

Dates
First available in Project Euclid: 29 August 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1188405622

Digital Object Identifier
doi:10.1214/009053606000001604

Mathematical Reviews number (MathSciNet)
MR2351097

Zentralblatt MATH identifier
1209.62082

Subjects
Primary: 62G05: Estimation 62G08: Nonparametric regression 62G20: Asymptotic properties

Keywords
dark matter empirical processes isotonic estimation least concave majorant regression function velocity dispersions

Citation

Wang, Xiao; Woodroofe, Michael. A Kiefer–Wolfowitz comparison theorem for Wicksell’s problem. Ann. Statist. 35 (2007), no. 4, 1559--1575. doi:10.1214/009053606000001604. https://projecteuclid.org/euclid.aos/1188405622


Export citation

References

  • Antoniadis, A., Fan, J. and Gijbels, I. (2001). A wavelet method for unfolding sphere size distributions. Canad. J. Statist. 29 251–268.
  • Binney, J. and Tremaine, S. (1987). Galactic Dynamics. Princeton Univ. Press, Princeton, NJ.
  • Chow, Y. S. and Teicher, H. (1997). Probability Theory: Independence, Interchangeability, Martingales, 3rd ed. Springer, New York.
  • Golubev, G. K. and Levit, B. Y. (1998). Asymptotically efficient estimation in the Wicksell problem. Ann. Statist. 26 2407–2419.
  • Groeneboom, P. and Jongbloed, G. (1995). Isotonic estimation and rates of convergence in Wicksell's problem. Ann. Statist. 23 1518–1542.
  • Hall, P. and Smith, R. L. (1988). The kernel method for unfolding sphere size distributions. J. Comput. Phys. 74 409–421.
  • Irwin, M. and Hatzidimitriou, D. (1995). Structural parameters for the galactic dwarf spheroidals. Monthly Notices Roy. Astronomical Soc. 277 1354–1378.
  • Kiefer, J. and Wolfowitz, J. (1976). Asymptotically minimax estimation of concave and convex distribution functions. Z. Wahrsch. Verw. Gebiete 34 73–85.
  • Robertson, T., Wright, F. T. and Dykstra, R. L. (1988). Order Restricted Statistical Inference. Wiley, Chichester.
  • van der Vaart, A. and Wellner, J. (1996). Weak Convergence and Empirical Processes. With Applications to Statistics. Springer, New York.
  • Wang, X., Woodroofe, M., Walker, M., Mateo, M. and Olszewski, E. (2005). Estimating dark matter distributions. Astrophysical J. 626 145–158.
  • Wicksell, S. D. (1925). The corpuscle problem. Biometrika 17 84–99.