On the minimizing point of the incorrectly centered empirical process and its limit distribution in nonregular experiments

Dietmar Ferger

ESAIM: Probability and Statistics (2005)

  • Volume: 9, page 307-322
  • ISSN: 1292-8100

Abstract

top
Let F n be the empirical distribution function (df) pertaining to independent random variables with continuous df F . We investigate the minimizing point τ ^ n of the empirical process F n - F 0 , where F 0 is another df which differs from F . If F and F 0 are locally Hölder-continuous of order α at a point τ our main result states that n 1 / α ( τ ^ n - τ ) converges in distribution. The limit variable is the almost sure unique minimizing point of a two-sided time-transformed homogeneous Poisson-process with a drift. The time-transformation and the drift-function are of the type | t | α .

How to cite

top

Ferger, Dietmar. "On the minimizing point of the incorrectly centered empirical process and its limit distribution in nonregular experiments." ESAIM: Probability and Statistics 9 (2005): 307-322. <http://eudml.org/doc/245175>.

@article{Ferger2005,
abstract = {Let $F_n$ be the empirical distribution function (df) pertaining to independent random variables with continuous df $F$. We investigate the minimizing point $\hat\{\tau \}_n$ of the empirical process $F_n - F_0$, where $F_0$ is another df which differs from $F$. If $F$ and $F_0$ are locally Hölder-continuous of order $\alpha $ at a point $\tau $ our main result states that $n^\{1/\alpha \}(\hat\{\tau \}_n - \tau )$ converges in distribution. The limit variable is the almost sure unique minimizing point of a two-sided time-transformed homogeneous Poisson-process with a drift. The time-transformation and the drift-function are of the type $|t|^\{\alpha \}$.},
author = {Ferger, Dietmar},
journal = {ESAIM: Probability and Statistics},
keywords = {rescaled empirical process; argmin-CMT; Poisson-process; weak convergence in $D(\mathbb \{R\})$; Rescaled empirical process; weak convergence in },
language = {eng},
pages = {307-322},
publisher = {EDP-Sciences},
title = {On the minimizing point of the incorrectly centered empirical process and its limit distribution in nonregular experiments},
url = {http://eudml.org/doc/245175},
volume = {9},
year = {2005},
}

TY - JOUR
AU - Ferger, Dietmar
TI - On the minimizing point of the incorrectly centered empirical process and its limit distribution in nonregular experiments
JO - ESAIM: Probability and Statistics
PY - 2005
PB - EDP-Sciences
VL - 9
SP - 307
EP - 322
AB - Let $F_n$ be the empirical distribution function (df) pertaining to independent random variables with continuous df $F$. We investigate the minimizing point $\hat{\tau }_n$ of the empirical process $F_n - F_0$, where $F_0$ is another df which differs from $F$. If $F$ and $F_0$ are locally Hölder-continuous of order $\alpha $ at a point $\tau $ our main result states that $n^{1/\alpha }(\hat{\tau }_n - \tau )$ converges in distribution. The limit variable is the almost sure unique minimizing point of a two-sided time-transformed homogeneous Poisson-process with a drift. The time-transformation and the drift-function are of the type $|t|^{\alpha }$.
LA - eng
KW - rescaled empirical process; argmin-CMT; Poisson-process; weak convergence in $D(\mathbb {R})$; Rescaled empirical process; weak convergence in
UR - http://eudml.org/doc/245175
ER -

References

top
  1. [1] P. Billingsley, Convergence of probability measures. Wiley, New York (1968). Zbl0172.21201MR233396
  2. [2] Z.W. Birnbaum and R. Pyke, On some distributions related to the statistic D n + . Ann. Math. Statist. 29 (1958) 179–187. Zbl0089.14803
  3. [3] Z.W. Birnbaum and F.H. Tingey, One-sided confidence contours for probability distribution functions. Ann. Math. Statist. 22 (1951) 592–596. Zbl0044.14601
  4. [4] F.P. Cantelli, Considerazioni sulla legge uniforme dei grandi numeri e sulla generalizzazione di un fondamentale teorema del sig. Paul Levy. Giorn. Ist. Ital. Attuari 4 (1933) 327–350. Zbl0007.21802
  5. [5] J. Donsker, Justification and extension of Doob’s heuristic approach to the Kolmogorov-Smirnov theorems. Ann. Math. Statist. 23 (1952) 277–281. Zbl0046.35103
  6. [6] R.M. Dudley, Weak convergence of probabilities on nonseparable metric spaces and empirical measures on Euclidean spaces. Illinois J. Math. 10 (1966) 109–126. Zbl0178.52502
  7. [7] R.M. Dudley, Measures on nonseparable metric spaces. Illinois J. Math. 11 (1967) 449–453. Zbl0152.24501
  8. [8] R.M. Dudley, Uniform central limit theorems. Cambridge University Press, New York (1999). Zbl0951.60033MR1720712
  9. [9] M. Dwass, On several statistics related to empirical distribution functions. Ann. Math. Statist. 29 (1958) 188–191. Zbl0089.14804
  10. [10] R. Dykstra and Ch. Carolan, The distribution of the argmax of two-sided Brownian motion with parabolic drift. J. Statist. Comput. Simul. 63 (1999) 47–58. Zbl0946.65001
  11. [11] D. Ferger, The Birnbaum-Pyke-Dwass theorem as a consequence of a simple rectangle probability. Theor. Probab. Math. Statist. 51 (1995) 155–157. Zbl0934.62017
  12. [12] D. Ferger, Analysis of change-point estimators under the null hypothesis. Bernoulli 7 (2001) 487–506. Zbl1006.62022
  13. [13] D. Ferger, A continuous mapping theorem for the argmax-functional in the non-unique case. Statistica Neerlandica 58 (2004) 83–96. Zbl1090.60032
  14. [14] D. Ferger, Cube root asymptotics for argmin-estimators. Unpublished manuscript, Technische Universität Dresden (2005). 
  15. [15] V. Glivenko, Sulla determinazione empirica delle leggi die probabilita. Giorn. Ist. Ital. Attuari 4 (1933) 92–99. Zbl0006.17403
  16. [16] P. Groneboom, Brownian motion with a parabolic drift and Airy Functions. Probab. Th. Rel. Fields 81 (1989) 79–109. 
  17. [17] P. Groneboom and J.A. Wellner, Computing Chernov’s distribution. J. Comput. Graphical Statist. 10 (2001) 388–400. 
  18. [18] J. Hoffman-Jørgensen, Stochastic processes on Polish spaces. (Published (1991): Various Publication Series No. 39, Matematisk Institut, Aarhus Universitet) (1984). Zbl0919.60003MR1217966
  19. [19] I.A. Ibragimov and R.Z. Has’minskii, Statistical Estimation: Asymptotic Theory. Springer-Verlag, New York (1981). Zbl0467.62026
  20. [20] O. Kallenberg, Foundations of Modern Probability. Springer-Verlag, New York (1999). Zbl0892.60001MR1876169
  21. [21] K. Knight, Epi-convergence in distribution and stochastic equi-semicontinuity. Technical Report, University of Toronto (1999) 1–22. 
  22. [22] A.N. Kolmogorov, Sulla determinazione empirica di una legge di distribuzione. Giorn. Ist. Ital. Attuari 4 (1933) 83–91. Zbl0006.17402
  23. [23] N.H. Kuiper, Alternative proof of a theorem of Birnbaum and Pyke. Ann. Math. Statist. 30 (1959) 251–252. Zbl0119.15003
  24. [24] T. Lindvall, Weak convergence of probability measures and random functions in the function space D [ 0 , ∞ ) . J. Appl. Prob. 10 (1973) 109–121. Zbl0258.60008
  25. [25] P. Massart, The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality. Ann. Probab. 18 (1990) 1269–1283. Zbl0713.62021
  26. [26] G.Ch. Pflug, On an argmax-distribution connected to the Poisson process, in Proc. of the fifth Prague Conference on asymptotic statistics, P. Mandl, H. Husková Eds. (1993) 123–130. 
  27. [27] G.R. Shorack and J.A. Wellner, Empirical processes with applications to statistics. Wiley, New York (1986). Zbl1170.62365MR838963
  28. [28] N.V. Smirnov, Näherungsgesetze der Verteilung von Zufallsveränderlichen von empirischen Daten. Usp. Mat. Nauk. 10 (1944) 179–206. Zbl0063.07087
  29. [29] L. Takács, Combinatorial Methods in the theory of stochastic processes. Robert E. Krieger Publishing Company, Huntingtun, New York (1967). Zbl0376.60016
  30. [30] A.W. van der Vaart and J.A. Wellner, Weak convergence of empirical processes. Springer-Verlag, New York (1996). Zbl0862.60002MR1385671

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.