Hypothesis of Randomness: Multidimensional Case

Conformity of the observed sequence with the randomness hypothesis is an important prerequisite for many statistical hypotheses. From the beginning of 1940's a variety of test for randomness hypothesis were developed. The research carried out to this day, one the one hand, resolves various theoretical questions, while on the other hand, it provides well-defined test statistics with known asymptotic properties. One of the randomness hypothesis conformity check subtask is so-called spatial randomness check. This subtask is devoted to checking whether sample of spatial points are uniformly distributed over the given support. In this paper a graph-based test for spatial randomness is investigated. The perspectives of its optimization and widening the application area are proposed. Experiments have shown that sparsing of the initial graph based on the Erdős-Rényi model negatively affects the result, and so is undesirable. It was also discovered that the test can be applied in the case of spatial sample being defined on the support different from unit hypercube. Moreover, an empirical conclusion was deduced that the test can be applied in case of non-euclidean metric being used, however, formal proof of this proposition is yet to be formulated.