On Estimating the Asymptotic Variance of Stationary Point Processes
Tóm tắt
We investigate a class of kernel estimators $\widehat{\sigma}^2_n$ of the asymptotic variance σ 2 of a d-dimensional stationary point process $\Psi = \sum_{i\ge 1}\delta_{X_i}$ which can be observed in a cubic sampling window $W_n = [-n,n]^d\,$ . σ 2 is defined by the asymptotic relation $Var(\Psi(W_n)) \sim \sigma^2 \,(2n)^d$ (as n → ∞) and its existence is guaranteed whenever the corresponding reduced covariance measure $\gamma^{(2)}_{red}(\cdot)$ has finite total variation. Depending on the rate of decay (polynomially or exponentially) of the total variation of $\gamma^{(2)}_{red}(\cdot)$ outside of an expanding ball centered at the origin, we determine optimal bandwidths b n (up to a constant) minimizing the mean squared error of $\widehat{\sigma}^2_n$ . The case when $\gamma^{(2)}_{red}(\cdot)$ has bounded support is of particular interest. Further we suggest an isotropised estimator $\widetilde{\sigma}^2_n$ suitable for motion-invariant point processes and compare its properties with $\widehat{\sigma}^2_n$ . Our theoretical results are illustrated and supported by a simulation study which compares the (relative) mean squared errors of $\widehat{\sigma}^2_n$ for planar Poisson, Poisson cluster, and hard-core point processes and for various values of n b n .
Tài liệu tham khảo
citation_journal_title=Aust N Z J Stat; citation_title=Kernel estimation of the spectral density of stationary random closed sets; citation_author=S Böhm, L Heinrich, V Schmidt; citation_volume=46; citation_publication_date=2004; citation_pages=41-52; citation_doi=10.1111/j.1467-842X.2004.00310.x; citation_id=CR1
citation_title=An introduction to the theory of point processes; citation_publication_date=1988; citation_id=CR2; citation_author=DJ Daley; citation_author=D Vere-Jones; citation_publisher=Springer
citation_journal_title=Math Methods Stat; citation_title=Normal approximation for some mean-value estimates of absolutely regular tessellations; citation_author=L Heinrich; citation_volume=3; citation_publication_date=1994; citation_pages=1-24; citation_id=CR3
Heinrich L (2008) Asymphotic goodness-of-fit tests for point processes based on scaled empirical K-functions (submitted)
citation_journal_title=Adv Appl Probab; citation_title=Normal convergence of multidimensional shot noise and rates of this convergence; citation_author=L Heinrich, V Schmidt; citation_volume=17; citation_publication_date=1985; citation_pages=709-730; citation_doi=10.2307/1427084; citation_id=CR4
citation_journal_title=J Nonparametr Stat; citation_title=Strong convergence of kernel estimators for product densities of absolutely regular point processes; citation_author=L Heinrich, E Liebscher; citation_volume=8; citation_publication_date=1997; citation_pages=65-96; citation_doi=10.1080/10485259708832715; citation_id=CR5
citation_title=Statistical analysis and modelling of spatial point patterns; citation_publication_date=2008; citation_id=CR6; citation_author=J Illian; citation_author=A Penttinen; citation_author=H Stoyan; citation_author=D Stoyan; citation_publisher=Wiley
citation_title=Central limit theorem and convergence of empirical processes of stationary point processes; citation_inbook_title=Point processes and queueing problems; citation_publication_date=1980; citation_pages=117-161; citation_id=CR7; citation_author=E Jolivet; citation_publisher=North-Holland
citation_journal_title=Adv Appl Probab; citation_title=Spectral theory for random closed sets and estimating the covariance via frequence space; citation_author=K Koch, J Ohser, K Schladitz; citation_volume=35; citation_publication_date=2003; citation_pages=603-613; citation_doi=10.1239/aap/1059486820; citation_id=CR8
citation_journal_title=J Appl Probab; citation_title=Asymptotic properties of stereological estimators for stationary random sets; citation_author=S Mase; citation_volume=19; citation_publication_date=1982; citation_pages=111-126; citation_doi=10.2307/3213921; citation_id=CR9
citation_title=Statistical analysis of microstructures in material science; citation_publication_date=2000; citation_id=CR10; citation_author=J Ohser; citation_author=F Mücklich; citation_publisher=Wiley
citation_journal_title=Biom J; citation_title=On the second-order and orientation analysis of planar stationary point processes; citation_author=J Ohser, D Stoyan; citation_volume=23; citation_publication_date=1981; citation_pages=523-533; citation_doi=10.1002/bimj.4710230602; citation_id=CR11
citation_title=Statistical inference for spatial processes; citation_publication_date=1988; citation_id=CR12; citation_author=BD Ripley; citation_publisher=Cambridge University Press
citation_journal_title=J R Stat Soc B; citation_title=Moment estimation for statistics from marked point processes; citation_author=DN Politis, M Sherman; citation_volume=63; citation_issue=2; citation_publication_date=2001; citation_pages=261-275; citation_doi=10.1111/1467-9868.00284; citation_id=CR13
citation_journal_title=Biometrika; citation_title=Asymptotically optimal estimation for the reduced second moment measure of point processes; citation_author=M Stein; citation_volume=80; citation_publication_date=1993; citation_pages=443-449; citation_doi=10.1093/biomet/80.2.443; citation_id=CR14
citation_journal_title=Scand J Statist; citation_title=Improving ratio estimators of second order point process characeristics; citation_author=D Stoyan, H Stoyan; citation_volume=27; citation_publication_date=2000; citation_pages=641-656; citation_doi=10.1111/1467-9469.00213; citation_id=CR15
citation_title=Stochastic geometry and its applications; citation_publication_date=1995; citation_id=CR16; citation_author=D Stoyan; citation_author=WS Kendall; citation_author=J Mecke; citation_publisher=Wiley
citation_title=An introduction to measure and probability; citation_publication_date=1997; citation_id=CR17; citation_author=JC Taylor; citation_publisher=Springer