Multiple testing has been a prominent topic in statistical research. Despite extensive work in this area, controlling false discoveries remains a challenging task, especially when the test statistics exhibit dependence. Various methods have been proposed to estimate the false discovery proportion (FDP) under arbitrary dependencies among the test statistics. One key approach is to transform arbitrary dependence into weak dependence and subsequently establish the strong consistency of FDP and false discovery rate under weak dependence. As a result, FDPs converge to the same asymptotic limit within the framework of weak dependence. However, we have observed that the asymptotic variance of FDP can be significantly influenced by the dependence structure of the test statistics, even when they exhibit only weak dependence. Quantifying this variability is of great practical importance, as it serves as an indicator of the quality of FDP estimation from the data. To the best of our knowledge, there is limited research on this aspect in the literature. In this paper, we aim to fill in this gap by quantifying the variation of FDP, assuming that the test statistics exhibit weak dependence and follow normal distributions. We begin by deriving the asymptotic expansion of the FDP and subsequently investigate how the asymptotic variance of the FDP is influenced by different dependence structures. Based on the insights gained from this study, we recommend that in multiple testing procedures utilizing FDP, reporting both the mean and variance estimates of FDP can provide a more comprehensive assessment of the study\'s outcomes.

A change point is a location or time at which observations or data obey two different models: before and after. In real problems, we may know some prior information about the location of the change point, say at the right or left tail of the sequence. How does one incorporate the prior information into the current cumulative sum (CUSUM) statistics? We propose a new class of weighted CUSUM statistics with three different types of quadratic weights accounting for different prior positions of the change points. One interpretation of the weights is the mean duration in a random walk. Under the normal model with known variance, the exact distributions of these statistics are explicitly expressed in terms of eigenvalues. Theoretical results about the explicit difference of the distributions are valuable. The expansions of asymptotic distributions are compared with the expansion of the limit distributions of the Cramér-von Mises statistic and the Anderson and Darling statistic. We provide some extensions from independent normal responses to more interesting models, such as graphical models, the mixture of normals, Poisson, and weakly dependent models. Simulations suggest that the proposed test statistics have better power than the graph-based statistics. We illustrate their application to a detection problem with video data.

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\'.