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Dependent phenomena, such as relational, spatial and temporal phenomena, tend to be characterized by local dependence in the sense that units which are close in a well‐defined sense are dependent. In contrast with spatial and temporal phenomena, though, relational phenomena tend to lack a natural neighbourhood structure in the sense that it is unknown which units are close and thus dependent. Owing to the challenge of characterizing local dependence and constructing random graph models with local dependence, many conventional exponential family random graph models induce strong dependence and are not amenable to statistical inference. We take first steps to characterize local dependence in random graph models, inspired by the notion of finite neighbourhoods in spatial statistics and M‐dependence in time series, and we show that local dependence endows random graph models with desirable properties which make them amenable to statistical inference. We show that random graph models with local dependence satisfy a natural domain consistency condition which every model should satisfy, but conventional exponential family random graph models do not satisfy. In addition, we establish a central limit theorem for random graph models with local dependence, which suggests that random graph models with local dependence are amenable to statistical inference. We discuss how random graph models with local dependence can be constructed by exploiting either observed or unobserved neighbourhood structure. In the absence of observed neighbourhood structure, we take a Bayesian view and express the uncertainty about the neighbourhood structure by specifying a prior on a set of suitable neighbourhood structures. We present simulation results and applications to two real world networks with ‘ground truth’.

We obtain explicit error bounds for the d-dimensional normal approximation on hyperrectangles for a random vector that has a Stein kernel, or admits an exchangeable pair coupling, or is a nonlinear statistic of independent random variables or a sum of n locally dependent random vectors.We assume the approximating normal distribution has a nonsingular covariance matrix. The error bounds vanish even when the dimension d is much larger than the sample size n. We prove our main results using the approach of Götze (1991) in Stein’s method, together with modifications of an estimate of Anderson, Hall and Titterington (1998) and a smoothing inequality of Bhattacharya and Rao (1976). For sums of n independent and identically distributed isotropic random vectors having a log-concave density, we obtain an error bound that is optimal up to a log n factor. We also discuss an application to multiple Wiener–Itô integrals.

The Berry-Esseen bound provides an upper bound on the Kolmogorov distance between a random variable and the normal distribution. In this paper, we establish Berry-Esseen bounds with optimal rates for self-normalized sums of locally dependent random variables, assuming only a second-moment condition. Our proof leverages Stein’s method and introduces a novel randomized concentration inequality, which may also be of independent interest for other applications. Our main results have applied to self-normalized sums of m -dependent random variables and graph dependency models.

Log-concavity of a joint survival function is proposed as a model for bivariate increasing failure rate (BIFR) distributions. Its connections with or distinctness from other notions of BIFR are discussed. A necessary and sufficient condition for a bivariate survival function to be log-concave (BIFR-LCC) is given that elucidates the impact of dependence between lifetimes on ageing. Illustrative examples are provided to explain BIFR-LCC for both positive and negative dependence.

A mantra often repeated in the introductory material to psychometrics and Item Response Theory (IRT) is that a Rasch model is a probabilistic version of a Guttman scale. The idea comes from the observation that a sigmoidal item response function provides a probabilistic version of the characteristic function that models an item response in the Guttman scale. It appears, however, more difficult to reconcile the assumption of local independence, which traditionally accompanies the Rasch model, with the item dependence existing in a Guttman scale. In recent work, an alternative probabilistic version of a Guttman scale was proposed, combining Knowledge Space Theory (KST) with IRT modeling, here referred to as KST-IRT. The present work has, therefore, a two-fold aim. Firstly, the estimation of the parameters involved in KST-IRT models is discussed. More in detail, two estimation methods based on the Expectation Maximization (EM) procedure are suggested, i.e., Marginal Maximum Likelihood (MML) and Gibbs sampling, and are compared on the basis of simulation studies. Secondly, for a Guttman scale, the estimates of the KST-IRT models are compared with those of the traditional combination of the Rasch model plus local independence under the interchange of the data generation processes. Results show that the KST-IRT approach might be more effective in capturing local dependence as it appears to be more robust under misspecification of the data generation process, but it comes with the price of an increased number of parameters.

The local independence assumption is crucial for the consistent estimation of item parameters in item response theory models. This article explores a pairwise likelihood estimation approach for the two-parameter logistic (2PL) model that treats the local dependence structure as a nuisance in the optimization function. Hence, item parameters can be consistently estimated without explicit modeling assumptions of the dependence structure. Two simulation studies demonstrate that the proposed pairwise likelihood estimation approach allows nearly unbiased and consistent item parameter estimation. Our proposed method performs similarly to the marginal maximum likelihood and pairwise likelihood estimation approaches, which also estimate the parameters for the local dependence structure.

Attribute reduction is a hot topic in the field of data mining. Compared with the traditional methods, the attribute reduction algorithm based on covering rough set is more suitable for dealing with numerical data. However, this kind of algorithm is still not efficient enough to deal with large-scale data. In this paper, we firstly propose ε-Boolean identification matrix of covering rough sets, give the calculation methods of dependence degree and local dependence degree, and further discuss their properties. Secondly, we give two attribute reduction algorithms based on dependence degree and local dependence degree, respectively. Finally, we test the performance of the algorithm through several UCI data sets. Experimental results show that the efficiency of our algorithm has been greatly improved. So it is more suitable for handling large-scale data processing problems, and can have wide application value.

This paper introduces two local conditional dependence matrices based on Spearman’s ρ and Kendall’s τ given the condition that the underlying random variables belong to the intervals determined by their quantiles. The robustness is studied by means of the influence functions of conditional Spearman’s ρ and Kendall’s τ . Using the two matrices, we construct the corresponding test statistics of local conditional dependence and derive their limit behavior including consistency, null and alternative asymptotic distributions. Simulation studies illustrate a superior power performance of the proposed Kendall-based test. Real data analysis with proposed methods provides a precise description and explanation of some financial phenomena in terms of mathematical statistics.

Item response theory (IRT) plays an important role in psychological and educational measurement. Unlike the classical testing theory, IRT models aggregate the item level information, yielding more accurate measurements. Most IRT models assume local independence, an assumption not likely to be satisfied in practice, especially when the number of items is large. Results in the literature and simulation studies in this paper reveal that misspecifying the local independence assumption may result in inaccurate measurements and differential item functioning. To provide more robust measurements, we propose an integrated approach by adding a graphical component to a multidimensional IRT model that can offset the effect of unknown local dependence. The new model contains a confirmatory latent variable component, which measures the targeted latent traits, and a graphical component, which captures the local dependence. An efficient proximal algorithm is proposed for the parameter estimation and structure learning of the local dependence. This approach can substantially improve the measurement, given no prior information on the local dependence structure. The model can be applied to measure both a unidimensional latent trait and multidimensional latent traits.

Item response theory (IRT) is one of the most widely utilized tools for item response analysis; however, local item and person independence, which is a critical assumption for IRT, is often violated in real testing situations. In this article, we propose a new type of analytical approach for item response data that does not require standard local independence assumptions. By adapting a latent space joint modeling approach, our proposed model can estimate pairwise distances to represent the item and person dependence structures, from which item and person clusters in latent spaces can be identified. We provide an empirical data analysis to illustrate an application of the proposed method. A simulation study is provided to evaluate the performance of the proposed method in comparison with existing methods.