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1. Hypothesis Testing for Hierarchical Structures in Cognitive Diagnosis Models

   https://doi.org/10.6339/21-JDS1024  

   Ma, Chenchen; Xu, Gongjun (January 2022, Journal of Data Science)

   Cognitive Diagnosis Models (CDMs) are a special family of discrete latent variable models widely used in educational, psychological and social sciences. In many applications of CDMs, certain hierarchical structures among the latent attributes are assumed by researchers to characterize their dependence structure. Specifically, a directed acyclic graph is used to specify hierarchical constraints on the allowable configurations of the discrete latent attributes. In this paper, we consider the important yet unaddressed problem of testing the existence of latent hierarchical structures in CDMs. We first introduce the concept of testability of hierarchical structures in CDMs and present sufficient conditions. Then we study the asymptotic behaviors of the likelihood ratio test (LRT) statistic, which is widely used for testing nested models. Due to the irregularity of the problem, the asymptotic distribution of LRT becomes nonstandard and tends to provide unsatisfactory finite sample performance under practical conditions. We provide statistical insights on such failures, and propose to use parametric bootstrap to perform the testing. We also demonstrate the effectiveness and superiority of parametric bootstrap for testing the latent hierarchies over non-parametric bootstrap and the naïve Chi-squared test through comprehensive simulations and an educational assessment dataset. 

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2. Sample Out-of-Sample Inference Based on Wasserstein Distance

   https://doi.org/10.1287/opre.2020.2028  

   Blanchet, Jose; Kang, Yang (May 2021, Operations Research)

   We present a novel inference approach that we call sample out-of-sample inference. The approach can be used widely, ranging from semisupervised learning to stress testing, and it is fundamental in the application of data-driven distributionally robust optimization. Our method enables measuring the impact of plausible out-of-sample scenarios in a given performance measure of interest, such as a financial loss. The methodology is inspired by empirical likelihood (EL), but we optimize the empirical Wasserstein distance (instead of the empirical likelihood) induced by observations. From a methodological standpoint, our analysis of the asymptotic behavior of the induced Wasserstein-distance profile function shows dramatic qualitative differences relative to EL. For instance, in contrast to EL, which typically yields chi-squared weak convergence limits, our asymptotic distributions are often not chi-squared. Also, the rates of convergence that we obtain have some dependence on the dimension in a nontrivial way but remain controlled as the dimension increases. 

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3. Testing high-dimensional multinomials with applications to text analysis

   https://doi.org/10.1093/jrsssb/qkae003  

   Cai, T Tony; Ke, Zheng T; Turner, Paxton (February 2024, Journal of the Royal Statistical Society Series B: Statistical Methodology)

   Abstract Motivated by applications in text mining and discrete distribution inference, we test for equality of probability mass functions of K groups of high-dimensional multinomial distributions. Special cases of this problem include global testing for topic models, two-sample testing in authorship attribution, and closeness testing for discrete distributions. A test statistic, which is shown to have an asymptotic standard normal distribution under the null hypothesis, is proposed. This parameter-free limiting null distribution holds true without requiring identical multinomial parameters within each group or equal group sizes. The optimal detection boundary for this testing problem is established, and the proposed test is shown to achieve this optimal detection boundary across the entire parameter space of interest. The proposed method is demonstrated in simulation studies and applied to analyse two real-world datasets to examine, respectively, variation among customer reviews of Amazon movies and the diversity of statistical paper abstracts. 

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4. On the Phase Transition of Wilk’s Phenomenon

   https://doi.org/10.1093/biomet/asaa078  

   He, Yinqiu; Meng, Bo; Zeng, Zhenghao; Xu, Gongjun (September 2020, Biometrika)

   null (Ed.)

   Abstract Wilk’s theorem, which offers universal chi-squared approximations for likelihood ratio tests, is widely used in many scientific hypothesis testing problems. For modern datasets with increasing dimension, researchers have found that the conventional Wilk’s phenomenon of the likelihood ratio test statistic often fails. Although new approximations have been proposed in high dimensional settings, there still lacks a clear statistical guideline regarding how to choose between the conventional and newly proposed approximations, especially for moderate-dimensional data. To address this issue, we develop the necessary and sufficient phase transition conditions for Wilk’s phenomenon under popular tests on multivariate mean and covariance structures. Moreover, we provide an in-depth analysis of the accuracy of chi-squared approximations by deriving their asymptotic biases. These results may provide helpful insights into the use of chi-squared approximations in scientific practices. 

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5. AUGUST: An Interpretable, Resolution-based Two-sample Test

   https://doi.org/10.51387/23-NEJSDS54  

   Brown, Benjamin; Zhang, Kai (December 2023, The New England Journal of Statistics in Data Science)

   Two-sample testing is a fundamental problem in statistics. While many powerful nonparametric methods exist for both the univariate and multivariate context, it is comparatively less common to see a framework for determining which data features lead to rejection of the null. In this paper, we propose a new nonparametric two-sample test named AUGUST, which incorporates a framework for interpretation while maintaining power comparable to existing methods. AUGUST tests for inequality in distribution up to a predetermined resolution using symmetry statistics from binary expansion. Designed for univariate and low to moderate-dimensional multivariate data, this construction allows us to understand distributional differences as a combination of fundamental orthogonal signals. Asymptotic theory for the test statistic facilitates p-value computation and power analysis, and an efficient algorithm enables computation on large data sets. In empirical studies, we show that our test has power comparable to that of popular existing methods, as well as greater power in some circumstances. We illustrate the interpretability of our method using NBA shooting data. 

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