Estimating Mark Functions Through Spectral Analysis for Marked Point Patterns

Abstract

Classical techniques for modeling numerical data associated to a regular grid have been widely developed in the literature. When a trigonometric model for the data is considered, it is possible to use the corresponding least squares (classical) estimators, but when the data are not observed on a regular grid, these estimators do not show appropriate properties. In this article we propose a novel way to model data that is not observed on a regular grid, and we establish a practical criterion, based on the mean squared error (MSE), to objectively decide which estimator should be used in each case: the inappropriate classical or the new unbiased estimator, which has greater variance. Jackknife and cross-validation techniques are used to follow a similar criterion in practice, when the MSE is not known. Finally, we present an application of the methodology to univariate and bivariate data.

Keywords:

• Cross-validation
• Jackknife
• Marked point pattern
• MSE
• Periodogram
• Thinning
• Trigonometric regression

Mathematics Subject Classification:

• 60G55
• 62M15
• 62M30

Acknowledgments

The refereee and editor are acknowledged with thanks as their comments have improved an earlier version of the article.

Work partially funded by grant MTM2004-06231 from the Spanish Ministry of Science and Culture, and CTBPRB/2002/201 from the Generalitat Valenciana.