Estimators of the asymptotic variance of stationary point processes - a comparison

Prokešová, Michaela. "Estimators of the asymptotic variance of stationary point processes - a comparison." Kybernetika 47.5 (2011): 678-695. <http://eudml.org/doc/196632>.

@article{Prokešová2011,
abstract = {We investigate estimators of the asymptotic variance $\sigma ^2$ of a $d$–dimensional stationary point process $\Psi $ which can be observed in convex and compact sampling window $W_n=n\, W$. Asymptotic variance of $\Psi $ is defined by the asymptotic relation $\{Var\}(\Psi (W_n)) \sim \sigma ^2 |W_n|$ (as $n \rightarrow \infty $) and its existence is guaranteed whenever the corresponding reduced covariance measure $\gamma ^\{(2)\}_\{\{\rm red\}\}(\cdot )$ has finite total variation. The three estimators discussed in the paper are the kernel estimator, the estimator based on the second order intesity of the point process and the subsampling estimator. We study the mean square consistency of the estimators. Since the expressions for the variance of the estimators are not available in closed form and depend on higher order moment measures of the point process, only the bias of the estimators can be compared theoretically. The second part of the paper is therefore devoted to a simulation study which compares the efficiency of the estimators by means of the mean squared error and for several clustered and repulsive point processes observed on middle-sized windows.},
author = {Prokešová, Michaela},
journal = {Kybernetika},
keywords = {reduced covariance measure; factorial moment and cumulant measures; kernel-type estimator; subsampling; mean squared error; Poisson cluster process; hard-core process; reduced covariance measure; factorial moment measures; cumulant measures; kernel-type estimator; subsampling; mean squared error; Poisson cluster process; hard-core process},
language = {eng},
number = {5},
pages = {678-695},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Estimators of the asymptotic variance of stationary point processes - a comparison},
url = {http://eudml.org/doc/196632},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Prokešová, Michaela
TI - Estimators of the asymptotic variance of stationary point processes - a comparison
JO - Kybernetika
PY - 2011
PB - Institute of Information Theory and Automation AS CR
VL - 47
IS - 5
SP - 678
EP - 695
AB - We investigate estimators of the asymptotic variance $\sigma ^2$ of a $d$–dimensional stationary point process $\Psi $ which can be observed in convex and compact sampling window $W_n=n\, W$. Asymptotic variance of $\Psi $ is defined by the asymptotic relation ${Var}(\Psi (W_n)) \sim \sigma ^2 |W_n|$ (as $n \rightarrow \infty $) and its existence is guaranteed whenever the corresponding reduced covariance measure $\gamma ^{(2)}_{{\rm red}}(\cdot )$ has finite total variation. The three estimators discussed in the paper are the kernel estimator, the estimator based on the second order intesity of the point process and the subsampling estimator. We study the mean square consistency of the estimators. Since the expressions for the variance of the estimators are not available in closed form and depend on higher order moment measures of the point process, only the bias of the estimators can be compared theoretically. The second part of the paper is therefore devoted to a simulation study which compares the efficiency of the estimators by means of the mean squared error and for several clustered and repulsive point processes observed on middle-sized windows.
LA - eng
KW - reduced covariance measure; factorial moment and cumulant measures; kernel-type estimator; subsampling; mean squared error; Poisson cluster process; hard-core process; reduced covariance measure; factorial moment measures; cumulant measures; kernel-type estimator; subsampling; mean squared error; Poisson cluster process; hard-core process
UR - http://eudml.org/doc/196632
ER -