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# Statistics and Its Interface

## Volume 9 (2016)

### Number 3

### Comparing the second-order properties of spatial point processes

Pages: 365 – 374

DOI: http://dx.doi.org/10.4310/SII.2016.v9.n3.a10

#### Authors

#### Abstract

Comparing the structural interactions of points in two independent stationary point patterns is as important as comparing the first-order properties or briefly their corresponding intensities. In the present study, three methods based on asymptotic distribution of the periodograms are proposed to test the equality of spectral densities of two independent stationary point processes. In the first method, the sample quantiles of the periodograms ratios are simply used and compared with the exact quantiles under the assumption of the equality of spectral densities. In the second method, a conditional likelihood ratio test is constructed, and the same idea of the first method is used to propose a Bayesian test for the ratio of the spectral density functions. The empirical powers and the empirical type I errors of the tests are also compared in a simulation study. The results emphasize the considerable powers of the tests and the empirical probability of type I errors is very close to the nominal level. Finally, the proposed methods are investigated by using two practical datasets: 1) comparing the locations of capillaries in healthy and cancerous prostate tissue sections and 2) comparing the locations of Alnus trees in two disjoint regions of Iran; both the regions in each dataset are almost of the same identical intensities. The same intensities leads to discovering of the treatment effect (cancer in the first data and location in the second data) in the second-order properties of point patterns.

#### Keywords

complete covariance density function, periodogram, second-order intensity function, spatial point process, spectral density function

#### 2010 Mathematics Subject Classification

Primary 60G55, 62M15. Secondary 62F15.