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1. With random regressors, least squares inference is robust to correlated errors with unknown correlation structure

   https://doi.org/10.1093/biomet/asae054  

   Zhang, Zifeng; Ding, Peng; Zhou, Wen; Wang, Haonan (January 2025, Biometrika)

   Abstract Linear regression is arguably the most widely used statistical method. With fixed regressors and correlated errors, the conventional wisdom is to modify the variance-covariance estimator to accommodate the known correlation structure of the errors. We depart from existing literature by showing that with random regressors, linear regression inference is robust to correlated errors with unknown correlation structure. The existing theoretical analyses for linear regression are no longer valid because even the asymptotic normality of the least squares coefficients breaks down in this regime. We first prove the asymptotic normality of the t statistics by establishing their Berry–Esseen bounds based on a novel probabilistic analysis of self-normalized statistics. We then study the local power of the corresponding t tests and show that, perhaps surprisingly, error correlation can even enhance power in the regime of weak signals. Overall, our results show that linear regression is applicable more broadly than the conventional theory suggests, and they further demonstrate the value of randomization for ensuring robustness of inference. 

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2. Quantile partially linear additive model for data with dropouts and an application to modeling cognitive decline

   https://doi.org/10.1002/sim.9745  

   Maidman, Adam; Wang, Lan; Zhou, Xiao‐Hua; Sherwood, Ben (July 2023, Statistics in Medicine)

   The National Alzheimer's Coordinating Center Uniform Data Set includes test results from a battery of cognitive exams. Motivated by the need to model the cognitive ability of low‐performing patients we create a composite score from ten tests and propose to model this score using a partially linear quantile regression model for longitudinal studies with non‐ignorable dropouts. Quantile regression allows for modeling non‐central tendencies. The partially linear model accommodates nonlinear relationships between some of the covariates and cognitive ability. The data set includes patients that leave the study prior to the conclusion. Ignoring such dropouts will result in biased estimates if the probability of dropout depends on the response. To handle this challenge, we propose a weighted quantile regression estimator where the weights are inversely proportional to the estimated probability a subject remains in the study. We prove that this weighted estimator is a consistent and efficient estimator of both linear and nonlinear effects. 

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3. Heavy-tailed distributions, correlations, kurtosis and Taylor’s Law of fluctuation scaling

   https://doi.org/10.1098/rspa.2020.0610  

   Cohen, Joel E.; Davis, Richard A.; Samorodnitsky, Gennady (December 2020, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   null (Ed.)

   Pillai & Meng (Pillai & Meng 2016 Ann. Stat. 44 , 2089–2097; p. 2091) speculated that ‘the dependence among [random variables, rvs] can be overwhelmed by the heaviness of their marginal tails ·· ·’. We give examples of statistical models that support this speculation. While under natural conditions the sample correlation of regularly varying (RV) rvs converges to a generally random limit, this limit is zero when the rvs are the reciprocals of powers greater than one of arbitrarily (but imperfectly) positively or negatively correlated normals. Surprisingly, the sample correlation of these RV rvs multiplied by the sample size has a limiting distribution on the negative half-line. We show that the asymptotic scaling of Taylor’s Law (a power-law variance function) for RV rvs is, up to a constant, the same for independent and identically distributed observations as for reciprocals of powers greater than one of arbitrarily (but imperfectly) positively correlated normals, whether those powers are the same or different. The correlations and heterogeneity do not affect the asymptotic scaling. We analyse the sample kurtosis of heavy-tailed data similarly. We show that the least-squares estimator of the slope in a linear model with heavy-tailed predictor and noise unexpectedly converges much faster than when they have finite variances. 

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4. Flexoelectricity in atomic monolayers from first principles

   https://doi.org/10.1039/D0NR07803D  

   Kumar, Shashikant; Codony, David; Arias, Irene; Suryanarayana, Phanish (January 2021, Nanoscale)

   We study the flexoelectric effect in fifty-four select atomic monolayers using ab initio Density Functional Theory (DFT). Specifically, considering representative materials from each of the Group III monochalcogenides, transition metal dichalcogenides (TMDs), Groups IV, III–V, and V monolayers, Group IV dichalcogenides, Group IV monochalcogenides, transition metal trichalcogenides (TMTs), and Group V chalcogenides, we perform symmetry-adapted DFT simulations to calculate transversal flexoelectric coefficients along the principal directions at practically relevant bending curvatures. We find that the materials demonstrate linear behavior and have similar coefficients along both principal directions, with values for TMTs being up to a factor of five larger than those of graphene. In addition, we find electronic origins for the flexoelectric effect, which increases with monolayer thickness, elastic modulus along the bending direction, and sum of polarizability of constituent atoms. 

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5. Common and Sidorenko Linear Equations

   https://doi.org/10.1093/qmath/haaa068  

   Fox, Jacob; Pham, Huy Tuan; Zhao, Yufei (January 2021, The Quarterly Journal of Mathematics)

   Abstract A linear equation with coefficients in $$\mathbb{F}_q$$ is common if the number of monochromatic solutions in any two-coloring of $$\mathbb{F}_q^{\,n}$$ is asymptotically (as $$n \to \infty$$) at least the number expected in a random two-coloring. The linear equation is Sidorenko if the number of solutions in any dense subset of $$\mathbb{F}_q^{\,n}$$ is asymptotically at least the number expected in a random set of the same density. In this paper, we characterize those linear equations which are common, and those which are Sidorenko. The main novelty is a construction based on choosing random Fourier coefficients that shows that certain linear equations do not have these properties. This solves problems posed in a paper of Saad and Wolf. 

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