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Goodness-of-fit tests for complete spatial randomness based on Minkowski functionals of binary images. (English) Zbl 1404.62045

The authors consider the problem to construct a test for complete spatial randomness (CSR) of spatial point pattern data in a given study area \( S \). They model the observed data by \[ \mathcal{P}_{\lambda} :=\left\{ X_{1},\ldots,X_{N_{\lambda}} \right\}, \] where \( \lambda >0 \), \( \left( X_{j} \right)_{j \geq 1} \) is a sequence of independent identically distributed random vectors taking values in \( S \), and \( N_{\lambda} \) is a nonnegative integer-valued random variable, independent of \( \left( X_{j} \right)_{j \geq 1} \), with a distribution that depends on some parameter \( \lambda >0 \). For \( \mathcal{P}_{\lambda} \) to be CSR the \( \left( X_{j} \right)_{j \geq 1} \) are uniformly distributed on \( S \), and \( \mathcal{P}_{\lambda} \) has a Poisson distribution with expectation \( \lambda \).
The authors’ abstract: “We propose a class of goodness-of-fit tests for complete spatial randomness (CSR). In contrast to standard tests, our procedure utilizes a transformation of the data to a binary image, which is then characterized by geometric functionals. Under a suitable limiting regime, we derive the asymptotic distribution of the test statistics under the null hypothesis and almost sure limits under certain alternatives. The new tests are computationally efficient, and simulations show that they are strong competitors to other tests of CSR. The tests are applied to a real data set in gamma-ray astronomy, and immediate extensions are presented to encourage further work.”

MSC:

62G10 Nonparametric hypothesis testing
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
62M30 Inference from spatial processes
62P35 Applications of statistics to physics
85A25 Radiative transfer in astronomy and astrophysics

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