On the minimizing point of the incorrectly centered empirical process and its limit distribution in nonregular experiments
ESAIM: Probability and Statistics, Tome 9 (2005) , pp. 307-322.

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 .

DOI : https://doi.org/10.1051/ps:2005014
Classification : 60E15,  60F05,  60F17,  62E20
Mots clés : rescaled empirical process, argmin-CMT, Poisson-process, weak convergence in D()
@article{PS_2005__9__307_0,
     author = {Ferger, Dietmar},
     title = {On the minimizing point of the incorrectly centered empirical process and its limit distribution in nonregular experiments},
     journal = {ESAIM: Probability and Statistics},
     pages = {307--322},
     publisher = {EDP-Sciences},
     volume = {9},
     year = {2005},
     doi = {10.1051/ps:2005014},
     zbl = {1136.60315},
     mrnumber = {2174873},
     language = {en},
     url = {archive.numdam.org/item/PS_2005__9__307_0/}
}
Ferger, Dietmar. On the minimizing point of the incorrectly centered empirical process and its limit distribution in nonregular experiments. ESAIM: Probability and Statistics, Tome 9 (2005) , pp. 307-322. doi : 10.1051/ps:2005014. http://archive.numdam.org/item/PS_2005__9__307_0/

[1] P. Billingsley, Convergence of probability measures. Wiley, New York (1968). | MR 233396 | Zbl 0172.21201

[2] Z.W. Birnbaum and R. Pyke, On some distributions related to the statistic D n + . Ann. Math. Statist. 29 (1958) 179-187. | Zbl 0089.14803

[3] Z.W. Birnbaum and F.H. Tingey, One-sided confidence contours for probability distribution functions. Ann. Math. Statist. 22 (1951) 592-596. | Zbl 0044.14601

[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. | Zbl 0007.21802

[5] J. Donsker, Justification and extension of Doob's heuristic approach to the Kolmogorov-Smirnov theorems. Ann. Math. Statist. 23 (1952) 277-281. | Zbl 0046.35103

[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. | Zbl 0178.52502

[7] R.M. Dudley, Measures on nonseparable metric spaces. Illinois J. Math. 11 (1967) 449-453. | Zbl 0152.24501

[8] R.M. Dudley, Uniform central limit theorems. Cambridge University Press, New York (1999). | MR 1720712 | Zbl 0951.60033

[9] M. Dwass, On several statistics related to empirical distribution functions. Ann. Math. Statist. 29 (1958) 188-191. | Zbl 0089.14804

[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. | Zbl 0946.65001

[11] D. Ferger, The Birnbaum-Pyke-Dwass theorem as a consequence of a simple rectangle probability. Theor. Probab. Math. Statist. 51 (1995) 155-157. | Zbl 0934.62017

[12] D. Ferger, Analysis of change-point estimators under the null hypothesis. Bernoulli 7 (2001) 487-506. | Zbl 1006.62022

[13] D. Ferger, A continuous mapping theorem for the argmax-functional in the non-unique case. Statistica Neerlandica 58 (2004) 83-96. | Zbl 1090.60032

[14] D. Ferger, Cube root asymptotics for argmin-estimators. Unpublished manuscript, Technische Universität Dresden (2005).

[15] V. Glivenko, Sulla determinazione empirica delle leggi die probabilita. Giorn. Ist. Ital. Attuari 4 (1933) 92-99. | Zbl 0006.17403

[16] P. Groneboom, Brownian motion with a parabolic drift and Airy Functions. Probab. Th. Rel. Fields 81 (1989) 79-109.

[17] P. Groneboom and J.A. Wellner, Computing Chernov's distribution. J. Comput. Graphical Statist. 10 (2001) 388-400.

[18] J. Hoffman-Jørgensen, Stochastic processes on Polish spaces. (Published (1991): Various Publication Series No. 39, Matematisk Institut, Aarhus Universitet) (1984). | MR 1217966 | Zbl 0919.60003

[19] I.A. Ibragimov and R.Z. Has'Minskii, Statistical Estimation: Asymptotic Theory. Springer-Verlag, New York (1981). | Zbl 0467.62026

[20] O. Kallenberg, Foundations of Modern Probability. Springer-Verlag, New York (1999). | MR 1876169 | Zbl 0892.60001

[21] K. Knight, Epi-convergence in distribution and stochastic equi-semicontinuity. Technical Report, University of Toronto (1999) 1-22.

[22] A.N. Kolmogorov, Sulla determinazione empirica di una legge di distribuzione. Giorn. Ist. Ital. Attuari 4 (1933) 83-91. | Zbl 0006.17402

[23] N.H. Kuiper, Alternative proof of a theorem of Birnbaum and Pyke. Ann. Math. Statist. 30 (1959) 251-252. | Zbl 0119.15003

[24] T. Lindvall, Weak convergence of probability measures and random functions in the function space D[0,). J. Appl. Prob. 10 (1973) 109-121. | Zbl 0258.60008

[25] P. Massart, The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality. Ann. Probab. 18 (1990) 1269-1283. | Zbl 0713.62021

[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] G.R. Shorack and J.A. Wellner, Empirical processes with applications to statistics. Wiley, New York (1986). | MR 838963

[28] N.V. Smirnov, Näherungsgesetze der Verteilung von Zufallsveränderlichen von empirischen Daten. Usp. Mat. Nauk. 10 (1944) 179-206. | Zbl 0063.07087

[29] L. Takács, Combinatorial Methods in the theory of stochastic processes. Robert E. Krieger Publishing Company, Huntingtun, New York (1967). | Zbl 0376.60016

[30] A.W. Van Der Vaart and J.A. Wellner, Weak convergence of empirical processes. Springer-Verlag, New York (1996). | MR 1385671