Stochastic Quasi-Likelihood for Case-Control Point Pattern Data

ABSTRACT

We propose a novel stochastic quasi-likelihood estimation procedure for case-control point processes. Quasi-likelihood for point processes depends on a certain optimal weight function and for the new method the weight function is stochastic since it depends on the control point pattern. The new procedure also provides a computationally efficient implementation of quasi-likelihood for univariate point processes in which case a synthetic control point process is simulated by the user. Under mild conditions, the proposed approach yields consistent and asymptotically normal parameter estimators. We further show that the estimators are optimal in the sense that the associated Godambe information is maximal within a wide class of estimating functions for case-control point processes. The effectiveness of the proposed method is further illustrated using extensive simulation studies and two data examples.

Keywords:

• Case-control data
• Godambe information
• Optimal estimating equations
• Point process

Supplementary Material

The supplementary materials contain the proofs of Theorems 1–4 and of Lemma 1.

Acknowledgments

The authors thank the editor, the associate editor and anonymous referees for their constructive comments that lead to substantial improvements of the article. The authors also thank Prof. Hansheng Wang and Mr. Yu Chen for their help in collecting the Beijing restaurant location data.

Additional information

Funding

Xu’s research was supported by Collaboration Grants for Mathematicians from the Simons Foundation (Award Number: 524205). Waagepetersen’s research was supported by The Danish Council for Independent Research-Natural Sciences, grant DFF-7014-00074 “Statistics for point processes in space and beyond”, and by the “Centre for Stochastic Geometry and Advanced Bioimaging”, funded by grant 8721 from the Villum Foundation. Guan’s research was supported by National Institutes of Health grant R01 CA169043. The BCI forest dynamics research project was made possible by National Science Foundation grants to Stephen P. Hubbell: DEB-0640386, DEB-0425651, DEB-0346488, DEB-0129874, DEB-00753102, DEB-9909347, DEB-9615226, DEB-9615226, DEB-9405933, DEB-9221033, DEB-9100058, DEB-8906869, DEB-8605042, DEB-8206992, DEB-7922197, support from the Center for Tropical Forest Science, the Smithsonian Tropical Research Institute, the John D. and Catherine T. MacArthur Foundation, the Mellon Foundation, the Celera Foundation, and numerous private individuals, and through the hard work of over 100 people from 10 countries over the past two decades. The plot project is part of the Center for Tropical Forest Science, a global network of large-scale demographic tree plots. The BCI soils data set were collected and analyzed by J. Dalling, R. John and K. Harms with support from NSF DEB021104, 021115, 0212284, 0212818 and OISE 0314581 and OISE 031458, STRI and CTFS.