Hostname: page-component-cb9f654ff-hn9fh Total loading time: 0 Render date: 2025-08-27T12:45:26.988Z Has data issue: false hasContentIssue false
  • English
  • Français

On the minimizing point of the incorrectlycentered empirical process and its limit distributionin nonregular experiments

Published online by Cambridge University Press:  15 November 2005

Dietmar Ferger
Dietmar Ferger*
Affiliation:
Department of Mathematics, Dresden University of Technology, Helmholtzstr. 10, 01062 Dresden, Germany; ferger@math.tu-dresden.de
Get access

Abstract

Let F n be the empirical distribution function (df) pertainingto independent random variables with continuous df F. Weinvestigate the minimizing point $\hat\tau_n$ of the empiricalprocess Fn - F0 , where F 0 is another df which differs fromF. If F and F 0 are locally Hölder-continuous of orderα at a point τ our main result states that $n^{1/\alpha}(\hat\tau_n - \tau)$ converges in distribution. Thelimit variable is the almost sure unique minimizing point of atwo-sided time-transformed homogeneous Poisson-process with adrift. The time-transformation and the drift-function are of thetype |t|α.

Keywords

Rescaled empirical processargmin-CMTPoisson-processweak convergence in $D(\mathbb{R})$ .

Information

Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 9 , February 2005 , pp. 307 - 322
Copyright
© EDP Sciences, SMAI, 2005

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

P. Billingsley, Convergence of probability measures. Wiley, New York (1968).
Birnbaum, Z.W. and Pyke, R., On some distributions related to the statistic $D_n^+$ . Ann. Math. Statist. 29 (1958) 179–187. CrossRef
Birnbaum, Z.W. and Tingey, F.H., One-sided confidence contours for probability distribution functions. Ann. Math. Statist. 22 (1951) 592–596. CrossRef
Cantelli, F.P., 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.
Donsker, J., Justification and extension of Doob's heuristic approach to the Kolmogorov-Smirnov theorems. Ann. Math. Statist. 23 (1952) 277–281. CrossRef
Dudley, R.M., Weak convergence of probabilities on nonseparable metric spaces and empirical measures on Euclidean spaces. Illinois J. Math. 10 (1966) 109–126.
Dudley, R.M., Measures on nonseparable metric spaces. Illinois J. Math. 11 (1967) 449–453.
R.M. Dudley, Uniform central limit theorems. Cambridge University Press, New York (1999).
Dwass, M., On several statistics related to empirical distribution functions. Ann. Math. Statist. 29 (1958) 188–191. CrossRef
Dykstra, R. and Carolan, Ch., The distribution of the argmax of two-sided Brownian motion with parabolic drift. J. Statist. Comput. Simul. 63 (1999) 47–58. CrossRef
Ferger, D., The Birnbaum-Pyke-Dwass theorem as a consequence of a simple rectangle probability. Theor. Probab. Math. Statist. 51 (1995) 155–157.
Ferger, D., Analysis of change-point estimators under the null hypothesis. Bernoulli 7 (2001) 487–506. CrossRef
Ferger, D., A continuous mapping theorem for the argmax-functional in the non-unique case. Statistica Neerlandica 58 (2004) 83–96. CrossRef
D. Ferger, Cube root asymptotics for argmin-estimators. Unpublished manuscript, Technische Universität Dresden (2005).
Glivenko, V., Sulla determinazione empirica delle leggi die probabilita. Giorn. Ist. Ital. Attuari 4 (1933) 92–99.
Groneboom, P., Brownian motion with a parabolic drift and Airy Functions. Probab. Th. Rel. Fields 81 (1989) 79–109.
Groneboom, P. and Wellner, J.A., Computing Chernov's distribution. J. Comput. Graphical Statist. 10 (2001) 388–400.
J. Hoffman-Jørgensen, Stochastic processes on Polish spaces. (Published (1991): Various Publication Series No. 39, Matematisk Institut, Aarhus Universitet) (1984).
I.A. Ibragimov and R.Z. Has'minskii, Statistical Estimation: Asymptotic Theory. Springer-Verlag, New York (1981).
O. Kallenberg, Foundations of Modern Probability. Springer-Verlag, New York (1999).
K. Knight, Epi-convergence in distribution and stochastic equi-semicontinuity. Technical Report, University of Toronto (1999) 1–22.
Kolmogorov, A.N., Sulla determinazione empirica di una legge di distribuzione. Giorn. Ist. Ital. Attuari 4 (1933) 83–91.
Kuiper, N.H., Alternative proof of a theorem of Birnbaum and Pyke. Ann. Math. Statist. 30 (1959) 251–252. CrossRef
Lindvall, T., Weak convergence of probability measures and random functions in the function space D[0,∞). J. Appl. Prob. 10 (1973) 109–121. CrossRef
Massart, P., The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality. Ann. Probab. 18 (1990) 1269–1283. CrossRef
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.
G.R. Shorack and J.A. Wellner, Empirical processes with applications to statistics. Wiley, New York (1986).
Smirnov, N.V., Näherungsgesetze der Verteilung von Zufallsveränderlichen von empirischen Daten. Usp. Mat. Nauk. 10 (1944) 179–206.
L. Takács, Combinatorial Methods in the theory of stochastic processes. Robert E. Krieger Publishing Company, Huntingtun, New York (1967).
A.W. van der Vaart and J.A. Wellner, Weak convergence of empirical processes. Springer-Verlag, New York (1996).