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

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

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## Abstract

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Let ${F}_{n}$ be the empirical distribution function (df) pertaining to independent random variables with continuous df $F$. We investigate the minimizing point ${\stackrel{^}{\tau }}_{n}$ 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 ${n}^{1/\alpha }\left({\stackrel{^}{\tau }}_{n}-\tau \right)$ 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 ${|t|}^{\alpha }$.

## How to cite

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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 -

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