Detecting structural changes in evolving networks is an important problem in sequential statistical inference. In many modern applications, interactions arise among groups rather than pairs, so hypergraphs can capture such higher-order relationships more accurately than ordinary graphs. In this paper, we study sequential detection of abrupt structural changes in dynamic hypergraphs under the hypergraph beta -model. We focus on a stylized setting in which the network is initially homogeneous and a denser sub-hypergraph emerges after a single unknown change-point. To address this problem, we develop a cumulative sum (CUSUM) procedure for the case where both pre-change and post-change parameters are known, and a window-limited CUSUM (WL-CUSUM) procedure for the more realistic case where the post-change parameter is unknown and must be estimated online. Both procedures are constructed from degree-sequence observations, which are sufficient statistics under the hypergraph beta -model and are especially attractive when the full hypergraph structure is difficult to observe or use in practice. We establish asymptotic properties of the proposed procedures and study the choice of the window size in the unknown-parameter setting. Numerical experiments and a real-data illustration demonstrate the performance and practical relevance of the proposed methods.
Assume that phi(& centerdot;) is a non-negative, continuously differentiable weight function and phi '(& centerdot;) is nondecreasing on [0,infinity), and let 0= 0) of dimension alpha >= 1 and starting at zero such that E[tau(mu)]
We examine the optimality properties of the Gini index estimator under complex survey design involving stratification, clustering, and sub-stratification. While Darku et al. (Econometrics, 8, 26, 2020) considered only stratification and clustering and did not provide theoretical guarantees, this study addresses these limitations by proposing two procedures, that is, a purely sequential method and a two-stage method. Under suitable regularity conditions, we establish uniform continuity in probability for the proposed estimator, thereby contributing to the development of random central limit theorems under sequential sampling frameworks. Furthermore, we show that the resulting procedures satisfy both asymptotic first-order efficiency and asymptotic consistency. Simulation results demonstrate that the proposed procedures achieve the desired optimality properties across diverse settings. The practical utility of the methodology is further illustrated through an empirical application using data collected by the National Sample Survey agency of India.
We prove two inequalities introduced in our prior study of the graphical shape of the number of bit strings with a given score under an interesting scoring system. Generating functions are used to establish the inequalities, which in turn imply two of the salient graphical features, uni-modality near the zero score and shape asymmetry for positive versus negative scores. One inequality provides a lower bound on the expected value of a discrete random variable with probabilities proportional to a product of two binomial coefficients. The other inequality states that the expected value with respect to near central binomial coefficients of other binomial coefficients lying on an oblique ray in Pascal's triangle exceeds the expected value along an adjacent parallel ray to its left.
In this paper, we consider the prediction problem of the future failure times based on observed Type-II right-censored data from Lehmann family of distribution under simple step stress life testing (SSLT) scenario. We explore various point predictors including best unbiased predictor (BUP), maximum likelihood predictor (MLP), and conditional median predictor (CMP) of the censored items. Furthermore, we extend our analysis to include the derivation of prediction intervals (PIs) using pivotal quantity method and highest conditional density (HCD) method. We conduct an extensive simulation study to evaluate the effectiveness of the point and interval predictors. The criteria employed for comparative evaluation include prediction bias and root mean square prediction error for point predictors, alongside average length and coverage probability for prediction intervals. Finally, we have analyzed two real life data sets for illustrative purposes. The first data set contains the failure times of 35 solar lighting devices from an SSLT experiment, while the second data set contains the lifetimes of nanocrystalline embedded high-k device data.
