# Non-parametric indices of dependence between components for inhomogeneous multivariate random measures and marked sets

@article{Lieshout2016NonparametricIO, title={Non-parametric indices of dependence between components for inhomogeneous multivariate random measures and marked sets}, author={M.N.M. van Lieshout}, journal={arXiv: Statistics Theory}, year={2016}, pages={985-1015} }

We propose new summary statistics to quantify the association between the components in coverage-reweighted moment stationary multivariate random sets and measures. They are defined in terms of the coverage-reweighted cumulant densities and extend classic functional statistics for stationary random closed sets. We study the relations between these statistics and evaluate them explicitly for a range of models. Unbiased estimators are given for all statistics and applied to simulated examples.

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#### References

SHOWING 1-10 OF 60 REFERENCES

Summary statistics for inhomogeneous marked point processes

- Mathematics
- 2014

We propose new summary statistics for intensity-reweighted moment stationary marked point processes with particular emphasis on discrete marks. The new statistics are based on the $$n$$n-point… Expand

Indices of dependence between types in multivariate point patterns

- Mathematics
- 1999

We propose new summary statistics quantifying several forms of dependence between points of different types in a multi-type spatial point pattern. These statistics are the multivariate counterparts… Expand

Bandwidth selection for kernel estimators of the spatial intensity function

- Mathematics
- 2016

We discuss and compare various approaches to the problem of bandwidth selection for kernel estimators of intensity functions of spatial point processes. We also propose a new method based on the… Expand

Random Marked Sets

- Mathematics
- Advances in Applied Probability
- 2012

We aim to link random fields and marked point processes, and, therefore, introduce a new class of stochastic processes which are defined on a random set in . Unlike for random fields, the mark… Expand

A non-model-based approach to bandwidth selection for kernel estimators of spatial intensity functions

- Mathematics
- 2018

We propose a new bandwidth selection method for kernel estimators of spatial point process intensity functions. The method is based on an optimality criterion motivated by the Campbell formula… Expand

Improving Ratio Estimators of Second Order Point Process Characteristics

- Mathematics
- 2000

Ripley's K function, the L function and the pair correlation function are important second order characteristics of spatial point processes. These functions are usually estimated by ratio estimators,… Expand

A J-function for inhomogeneous spatio-temporal point processes

- Mathematics
- 2013

We propose a new summary statistic for inhomogeneous intensity-reweighted moment stationarity spatio-temporal point processes. The statistic is deﬁned in terms of the $n$-point correlation functions… Expand

A J-function for inhomogeneous point processes

- Mathematics
- 2010

We propose new summary statistics for intensity-reweighted moment stationary point processes, that is, point processes with translation invariant n-point correlation functions for all n ∈ N, that… Expand

The method of moments for random measures

- Mathematics
- 1983

SummaryFirst a general condition is derived which assures uniqueness of a random measure in terms of its moment measures. Then weak convergence of a sequence of random measures is established under… Expand

Testing the random field model hypothesis for random marked closed sets

- Mathematics
- 2016

Abstract When developing statistical models, it is of fundamental importance to decide if the various components are independent of one another, preferably using a formal statistical test.… Expand