### Abstract

Summary: Fitting regression models for intensity functions of spatial point processes is of great interest in ecological and epidemiological studies of association between spatially referenced events and geographical or environmental covariates. When Cox or cluster process models are used to accommodate clustering that is not accounted for by the available covariates, likelihoodbased inference becomes computationally cumbersome owing to the complicated nature of the likelihood function and the associated score function. It is therefore of interest to consider alternative, more easily computable estimating functions. We derive the optimal estimating function in a class of first-order estimating functions. The optimal estimating function depends on the solution of a certain Fredholm integral equation which in practice is solved numerically. The derivation of the optimal estimating function has close similarities to the derivation of quasi-likelihood for standard data sets. The approximate solution is further equivalent to a quasi-likelihood score for binary spatial data. We therefore use the term quasi-likelihood for our optimal estimating function approach. We demonstrate in a simulation study and a data example that our quasi-likelihood method for spatial point processes is both statistically and computationally efficient.

Original language | English (US) |
---|---|

Pages (from-to) | 677-697 |

Number of pages | 21 |

Journal | Journal of the Royal Statistical Society. Series B: Statistical Methodology |

Volume | 77 |

Issue number | 3 |

DOIs | |

State | Published - Jun 1 2015 |

### Fingerprint

### Keywords

- Estimating function
- Fredholm integral equation
- Godambe information
- Intensity function
- Regression model
- Spatial point process

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Journal of the Royal Statistical Society. Series B: Statistical Methodology*,

*77*(3), 677-697. https://doi.org/10.1111/rssb.12083

**Quasi-likelihood for spatial point processes.** / Guan, Yongtao; Jalilian, Abdollah; Waagepetersen, Rasmus.

Research output: Contribution to journal › Article

*Journal of the Royal Statistical Society. Series B: Statistical Methodology*, vol. 77, no. 3, pp. 677-697. https://doi.org/10.1111/rssb.12083

}

TY - JOUR

T1 - Quasi-likelihood for spatial point processes

AU - Guan, Yongtao

AU - Jalilian, Abdollah

AU - Waagepetersen, Rasmus

PY - 2015/6/1

Y1 - 2015/6/1

N2 - Summary: Fitting regression models for intensity functions of spatial point processes is of great interest in ecological and epidemiological studies of association between spatially referenced events and geographical or environmental covariates. When Cox or cluster process models are used to accommodate clustering that is not accounted for by the available covariates, likelihoodbased inference becomes computationally cumbersome owing to the complicated nature of the likelihood function and the associated score function. It is therefore of interest to consider alternative, more easily computable estimating functions. We derive the optimal estimating function in a class of first-order estimating functions. The optimal estimating function depends on the solution of a certain Fredholm integral equation which in practice is solved numerically. The derivation of the optimal estimating function has close similarities to the derivation of quasi-likelihood for standard data sets. The approximate solution is further equivalent to a quasi-likelihood score for binary spatial data. We therefore use the term quasi-likelihood for our optimal estimating function approach. We demonstrate in a simulation study and a data example that our quasi-likelihood method for spatial point processes is both statistically and computationally efficient.

AB - Summary: Fitting regression models for intensity functions of spatial point processes is of great interest in ecological and epidemiological studies of association between spatially referenced events and geographical or environmental covariates. When Cox or cluster process models are used to accommodate clustering that is not accounted for by the available covariates, likelihoodbased inference becomes computationally cumbersome owing to the complicated nature of the likelihood function and the associated score function. It is therefore of interest to consider alternative, more easily computable estimating functions. We derive the optimal estimating function in a class of first-order estimating functions. The optimal estimating function depends on the solution of a certain Fredholm integral equation which in practice is solved numerically. The derivation of the optimal estimating function has close similarities to the derivation of quasi-likelihood for standard data sets. The approximate solution is further equivalent to a quasi-likelihood score for binary spatial data. We therefore use the term quasi-likelihood for our optimal estimating function approach. We demonstrate in a simulation study and a data example that our quasi-likelihood method for spatial point processes is both statistically and computationally efficient.

KW - Estimating function

KW - Fredholm integral equation

KW - Godambe information

KW - Intensity function

KW - Regression model

KW - Spatial point process

UR - http://www.scopus.com/inward/record.url?scp=84929155185&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84929155185&partnerID=8YFLogxK

U2 - 10.1111/rssb.12083

DO - 10.1111/rssb.12083

M3 - Article

VL - 77

SP - 677

EP - 697

JO - Journal of the Royal Statistical Society. Series B: Statistical Methodology

JF - Journal of the Royal Statistical Society. Series B: Statistical Methodology

SN - 1369-7412

IS - 3

ER -