Hypotheses testing on infinite random graphs

Drawing on some recent results that provide the formalism necessary to definite stationarity for infinite random graphs, this paper initiates the study of statistical and learning questions pertaining to these objects. Specifically, a criterion for the existence of a consistent test for complex hypotheses is presented, generalizing the corresponding results on time series. As an application, it is shown how one can test that a tree has the Markov property, or, more generally, to estimate its memory

Goodness of fit tests for random multigraph models

Goodness of fit tests for two probabilistic multigraph models are presented. The first model is random stub matching given fixed degrees (RSM) so that edge assignments to vertex pair sites are dependent, and the second is independent edge assignments (IEA) according to a common probability distribution. Tests are performed using goodness of fit measures between the edge multiplicity sequence of an observed multigraph, and the expected one according to a simple or composite hypothesis. Test statistics of Pearson type and of likelihood ratio type are used, and the expected values of the Pearson statistic under the different models are derived. Test performances based on simulations indicate that even for small number of edges, the null distributions of both statistics are well approximated by their asymptotic χ2-distribution. The non-null distributions of the test statistics can be well approximated by proposed adjusted χ2-distributions used for power approximations. The influence of RSM on both test statistics is substantial for small number of edges and implies a shift of their distributions towards smaller values compared to what holds true for the null distributions under IEA. Two applications on social networks are included to illustrate how the tests can guide in the analysis of social structure

Asymptotic nonparametric statistical analysis of stationary time series

Stationarity is a very general, qualitative assumption, that can be assessed on the basis of application specifics. It is thus a rather attractive assumption to base statistical analysis on, especially for problems for which less general qualitative assumptions, such as independence or finite memory, clearly fail. However, it has long been considered too general to allow for statistical inference to be made. One of the reasons for this is that rates of convergence, even of frequencies to the mean, are not available under this assumption alone. Recently, it has been shown that, while some natural and simple problems such as homogeneity, are indeed provably impossible to solve if one only assumes that the data is stationary (or stationary ergodic), many others can be solved using rather simple and intuitive algorithms. The latter problems include clustering and change point estimation. In this volume I summarize these results. The emphasis is on asymptotic consistency, since this the strongest property one can obtain assuming stationarity alone. While for most of the problems for which a solution is found this solution is algorithmically realizable, the main objective in this area of research, the objective which is only partially attained, is to understand what is possible and what is not possible to do for stationary time series. The considered problems include homogeneity testing, clustering with respect to distribution, clustering with respect to independence, change-point estimation, identity testing, and the general question of composite hypotheses testing. For the latter problem, a topological criterion for the existence of a consistent test is presented. In addition, several open questions are discussed.Comment: This is the author's version of the homonymous volume published by Springer. The final authenticated version is available online at: https://doi.org/10.1007/978-3-030-12564-6 Further updates and corrections may be made her

Network dependence

I am grateful for funding from the Spanish Ministry of Economy and Competitiveness (MDM2014-0431 and ECO2017-86675-P) and the Community of Madrid (MadEco-CM S2015/HUM-3444)Programa de Doctorado en Economía por la Universidad Carlos III de MadridPresidente: Wenceslao González Manteiga.- Secretario: Carlos Velasco Gómez.- Vocal: Gábor Lugos