Goodness-of-fit tests for spatial point processes: A review

Goodness-of-fit tests based on the empirical Wasserstein distance are proposed for simple and composite null hypotheses involving general multivariate distributions. For group families, the procedure is to be implemented after preliminary reduction of the data via invariance. This property allows for calculation of exact critical values and p-values at finite sample sizes. Applications include testing for location–scale families and testing for families arising from affine transformations, such as elliptical distributions with given standard radial density and unspecified location vector and scatter matrix. A novel test for multivariate normality with unspecified mean vector and covariance matrix arises as a special case. For more general parametric families, we propose a parametric bootstrap procedure to calculate critical values. The lack of asymptotic distribution theory for the empirical Wasserstein distance means that the validity of the parametric bootstrap under the null hypothe …

We consider marked empirical processes indexed by a randomly projected functional covariate to construct goodness-of-fit tests for the functional linear model with scalar response. The test statistics are built from continuous functionals over the projected process, resulting in computationally efficient tests that exhibit root-$n$ convergence rates and circumvent the curse of dimensionality. The weak convergence of the empirical process is obtained conditionally on a random direction, whilst the almost surely equivalence between the testing for significance expressed on the original and on the projected functional covariate is proved. The computation of the test in practice involves calibration by wild bootstrap resampling and the combination of several $p$-values, arising from different projections, by means of the false discovery rate method. The finite sample properties of the tests are illustrated in a simulation study for a variety of linear models, underlying processes, and alte …

This result suggests discrepancy measures between empirical and theoretical K K -function with known limit distribution which allow to perform goodness-of-fit tests for checking a hypothesized point process based only on its intensity and (generalized) K K -function. Similar test statistics are derived for testing the hypothesis that two independent point processes in W n W_{n} have the same distribution without explicit knowledge of their intensities and K K -functions.

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.

Summary The inspection of residuals is a fundamental step for investigating the quality of adjustment of a parametric model to data. For spatial point processes, the concept of residuals has been recently proposed as an empirical counterpart of the Campbell equilibrium equation for marked Gibbs point processes. The paper focuses on stationary marked Gibbs point processes and deals with asymptotic properties of residuals for such processes. In particular, the consistency and the asymptotic normality are obtained for a wide class of residuals including the classical residuals (raw, inverse and Pearson). On the basis of these asymptotic results, we define goodness-of-fit tests with type I error theoretically controlled. One of these tests constitutes an extension of the quadrat counting test that is widely used to test the null hypothesis of a homogeneous Poisson point process.