Estimating the K-function of a point process with an application to cosmology

The Annals of Statistics (Impact Factor: 2.53). 07/2000; DOI: 10.1214/aos/1015957468
Source: arXiv

ABSTRACT Motivated by the study of an important data set for understanding the large-scale structure of the universe, this work considers the estimation of the reduced second moment function, or K-function, of a stationary point process observed over a large number of segments of possibly varying lengths. Theory and simulation are used to compare the behavior of isotropic and rigid motion correction estimators and some modifications of these estimators. These results generally support the use of modified versions of the rigid motion correction. When applied to a catalog of astronomical objects known as absorbers, the proposed methods confirm results from earlier analyses of the absorber catalog showing clear evidence of clustering up to 50 Mpc and marginal evidence for clustering of matter on spatial scales beyond 100 Mpc, which is beyond the distance at which clustering of matter is now generally accepted to exist.

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    ABSTRACT: In this paper, we focus on resampling non-stationary weakly dependent point processes in two dimensions to make inference on the inhomogeneous K function (Baddeley et al., 2000). We provide theoretical results that show a consistency result of the bootstrap estimates of the variance as the observation region and resampling blocks increase in size. We present results of a simulation study that examines the performance of nominal 95% confidence intervals for the inhomogeneous K function obtained via our bootstrap procedure. The procedure is also applied to a rainforest dataset.
    Journal of Statistical Planning and Inference. 01/2010;
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    ABSTRACT: This volume is inspired by the many contributions of Arthur Getis to the field of spatial analysis. In 2004, Arthur Getis formally retired as the Stephen and Mary Birch Foundation Chair of Geographical Studies in the Department of Geography at San Diego State University. That transition to emeritus status marked the end of a magnificent career spanning more than four decades. It started with undergraduate education in geography at Pennsylvania State University, followed by a PhD from the University of Washington in 1961. At Washington, he was part of the generation that initiated the “quantitative revolution” in geography under the tutelage of William Garrison. His graduate cohort included, among others, Brian Berry, Waldo Tobler, Duane Marble, John Nystuen, Richard Morrill and William Bunge. His academic appointments started with a position at Michigan State University, after which he moved to Rutgers University. He went on to become head of the Geography Department at the University of Illinois in 1977, and joined San Diego State University in 1989. In addition, he held many visiting scholar appointments at leading international institutions, including Cambridge University and the University of Bristol in the UK and the University of California, Santa Barbara and Harvard University in the USA. During his career, Arthur Getis was awarded several honors and distinctions, such as the 1995 Albert Johnson Research Lecture at San Diego State University (captured in Getis, 1995c), the Walter Isard Award from the North American Regional Science Council (1997), the Robinson Lecture at The Ohio State University (1999), and the 2002 Distinguished Scholarship Award from the Association of American Geographers (AAG). In 2005, he was elected Fellow of the Regional Science Association International. He served as president of the Western Regional Science Association (1999) and of the University Consortium of Geographic Information Science (2002).
    12/2009: pages 1-20;
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    ABSTRACT: We focus on selecting optimal bandwidths for non-parametric estimation of the two-point correlation function of a point pattern. We obtain these optimal bandwidths by using a bootstrap approach to select a bandwidth that minimizes the integrated squared error. The variance term is estimated by using a non-parametric spatial bootstrap, whereas the bias term is estimated with a plug-in approach using a pilot estimator of the two-point correlation function based on a parametric model. The choice of parametric model for the pilot estimator is very flexible. Depending on applications, parametric statistical point models, physical models or functional models can be used. We also explore the use of the procedure for selecting adaptive optimal bandwidths. We investigate the performance of the bandwidth selection procedure by using a simulation study. In our data example, we apply our method to a Sloan Digital Sky Survey galaxy cluster catalogue by using a pilot estimator based on the power law functional model in cosmology. The resulting non-parametric two-point correlation function estimate is then used to estimate a cosmological mass bias parameter that describes the relationship between the galaxy mass distribution and the underlying matter distribution. Copyright (c) 2010 Royal Statistical Society.
    Applied Statistics 01/2010; 59(5):761-779. · 1.25 Impact Factor

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