Probabilistic integration of a continuous dynamical system is a way of systematically introducing discretisation error, at scales no larger than errors introduced by standard numerical discretisation, in order to enable thorough exploration of possible responses of the system to inputs. It is thus a potentially useful approach in a number of applications such as forward uncertainty quantification, inverse problems, and data assimilation. We extend the convergence analysis of probabilistic integrators for deterministic ordinary differential equations, as proposed by Conrad et al. (Stat Comput 27(4):1065–1082, 2017. https://doi.org/10.1007/s11222-016-9671-0), to establish mean-square convergence in the uniform norm on discrete- or continuous-time solutions under relaxed regularity assumptions on the driving vector fields and their induced flows. Specifically, we show that randomised high-order integrators for globally Lipschitz flows and randomised Euler integrators for dissipative vector fields with polynomially bounded local Lipschitz constants all have the same mean-square convergence rate as their deterministic counterparts, provided that the variance of the integration noise is not of higher order than the corresponding deterministic integrator. These and similar results are proven for probabilistic integrators where the random perturbations may be state-dependent, non-Gaussian, or non-centred random variables.
The constrict claim that ethnic heterogeneity drives down social trust has been empirically tested across the globe. Meta-analyses suggest that neighbourhood ethnic heterogeneity generally undermines ties within the neighbourhood (such as trust in neighbours), but concurrently has an inconsistent or even positive effect on interethnic ties (such as outgroup trust). While the composition of the living environment thus often seems to matter, when and where remain unclear. We contribute to the literature by: (1) scrutinizing the extent to which ethnic heterogeneity drives down trust in coethnic neighbours, non-coethnic neighbours, unknown neighbours and unknown non-neighbours similarly; (2) comparing effects of heterogeneity aggregated to geographical areas that vary in scale and type of boundary; and (3) assessing whether the impact of heterogeneity of the local area depends on the wider geographic context. We test our hypotheses on the Religion in Dutch Society 2011–2012 dataset, supplemented with uniquely detailed GIS-data of Statistics Netherlands. Our dependent variables are four different so-called wallet-items, which we model through spatial and multilevel regression techniques. We demonstrate that both trust in non-coethnic and coethnic neighbours are lower in heterogeneous environments. Trust in people outside the neighbourhood is not affected by local heterogeneity. Measures of heterogeneity aggregated to relatively large scales, such as, administrative municipalities and egohoods with a 4000 m radius, demonstrate the strongest negative relationships with our trust indicators. 相似文献
A class of asymptotically nonparametric test with contains a test proposed by Wei(1980), is considered for testing the equality of two continuous distribution funcitons when paired observations are subject to arbitrary right censorship. It is shown that under the null hypothesis each test statistic converges in distribution to the standard normal random variable. Furthermore. the Monte Carlo simulation results indicate that some tests in this class are more powerful than Wei's test. A generalization to incomplete censored paired data is also included. 相似文献