Enumeration of rooted binary perfect phylogenies.

This article proposes and studies two Huber-type estimation approaches, namely, the Huber instrumental variable (IV) estimation and the Huber generalized method of moments (GMM) estimation, for a spatial autoregressive model. We establish the consistency, asymptotic distributions, finite sample breakdown points, and influence functions of these estimators. Simulation studies show that compared to the corresponding traditional estimators (the two-stage least squares estimator, the best IV estimator, and the GMM estimator), our estimators are more robust when the unknown disturbances are long-tailed, and our estimators only lose a little efficiency when the disturbances are short-tailed. Moreover, the Huber GMM estimator also outperforms several robust estimators in the literature. Finally, we apply our estimation method to investigate the impact of the urban heat island effect on housing prices. A package is published on GitHub for practitioners to use in their empirical studies.

{Asymptotic theory, influence diagnostics and Monte Carlo simulations are used to demonstrate the robustness and efficiency of pretest and shrinkage estimators in semiparametric censored regression models.} This paper introduces shrinkage estimators namely, the preliminary test estimator, Stein-type estimator, and positive-part Stein estimator for semiparametric censored regression models. We derive the asymptotic distributions of the estimators and the nonparametric component, along with their asymptotic mean square error (AMSE) matrices. A novel contribution is the extension of Cook's distance as an influence diagnostic tool for these estimators, using a case-deletion approach and weighting by the respective AMSE matrices. Simulation studies assess estimator risk across different censoring levels and evaluate the proposed diagnostics. The methodology is applied to lung cancer data to demonstrate its practical utility.

The underlying distributions of random errors play an essential role in statistical inferences of regression models. The goodness-of-fit test on errors is especially important in high-dimensional regression settings because error distributions can also influence model selections. In this paper, we consider the goodness-of-fit test on errors in high-dimensional linear regression models. Under suitable assumptions of model sparsity and a sure screening property, we show that the Bickel-Rosenblatt-type test statistic, based on residuals from the refitted cross-validation procedure, has an asymptotic normal distribution, both under the null hypothesis and a fixed alternative.

We propose a novel goodness-of-fit test for the multivariate logistic distribution, leveraging the properties of its characteristic function. The test statistic is constructed by measuring a weighted L 2 -distance between the empirical characteristic function of the standardized data and the theoretical characteristic function of the standard multivariate logistic distribution. The approach is grounded in a radial Fourier transform, expressed via the Hankel transform, and yields an explicit, closed-form expression involving three integrals. We derive the asymptotic distribution of the test statistic under both the null hypothesis and contiguous alternatives, and prove its affine invariance. Critical values are determined using Monte Carlo simulation. Through extensive simulations, we demonstrate that the proposed test has excellent control of Type I error and exhibits superior power relative to existing methods, including the multivariate Kolmogorov–Smirnov and energy tests, especially in higher dimensions or under heavy-tailed alternatives. Applications to real data further underscore the test's practical utility and robustness. MATLAB and R implementations are made publicly available.

Abstract Correlation tests are very important tools for the pathway analysis or graphical modeling of high-dimensional data. In this study, we consider a correlation test under the strongly spiked eigenvalue (SSE) model in high-dimension, low-sample-size scenarios, where the sample size is much smaller than the dimension. High-dimensional data often fit the SSE model. Previously, a high-dimensional test for a correlation matrix under the non-SSE model was constructed using the extended cross-data-matrix (ECDM) methodology. Here, we show that an asymptotic distribution of the test statistic using ECDM methodology under the SSE model can be written using the distribution of the sum of some weighted chi-squared variables when both dimension and sample size reach infinity. We propose a new test procedure using the asymptotic distribution. We also show that the proposed test procedure has preferable properties for power and size. We discuss the performance of the test procedure through simulations and present a demonstration with actual data analyses using a microarray data set.

Mainstream random-effects meta-analysis of clinical trials (IVW, weights inversely proportional to the estimated variance), as has been shown in two publications, cannot be trusted. This is especially disturbing as meta-analysis, the combination of like studies to produce an overall estimate of effect size, is at the apex of most evidence pyramids. The asymptotic distribution theory of the mainstream fails because weighted linear combination theory can only be applied to weights that are constants with high degrees of accuracy. In fact, they are certainly volatile random variables. The asymptotic setting requires that the number of studies being combined goes to infinity. Our novel finding is that the coverage of their “95% confidence interval” for overall effect size converges to zero (strong inconsistency). We have conducted reanalysis of about 30 highly influential meta-analysis of clinical trials and found that the mainstream methods had unsupportable qualitative conclusions in about 10%, leading to scientifically unsupportable public health policies. Further, about another 15% had unsupportable quantitative conclusions. The two references offer an asymptotically valid alternative, based on ratio estimation borrowed from survey sampling methods. We also show that if you adjust for the bias of the IVW estimate, you obtain the unpopular equally weighted estimate.

ABSTRACT Fama and French (2015, 2017) introduce the five‐factor asset pricing model in the former paper and test their model on data from international financial markets in the latter paper. Each paper tests whether the five‐factor model represents returns by way of the Gibbons, Ross and Shanken (1989) (hereafter GRS) statistic. That statistic's null hypothesis jointly sets all cross‐section intercepts (alpha) to zero. The GRS statistic developed and presented in equation (4) on page 1124 of GRS (1989) is a cross‐section test of the one‐factor capital asset pricing model. Using the same data as Fama and French (2015, 2017), we show that the latter authors did not use the GRS (1989) statistic given in equation (4) on page 1124. In fact, they used a version of that statistic appropriate for the five‐factor model. To provide clarity on this issue, this paper provides a detailed mathematical derivation of the cross‐sectional variance of the OLS estimators of the intercepts when N versions of the K ‐factor model are estimated. This variance is then used to construct the enhanced version of the GRS statistic. Its finite sample distribution is then rigorously established. To obtain that distribution, restrictions are made on cross‐sectional variances and covariances of the errors of pricing models that are inconsistent with times series data. We derive the variance–covariance of the estimated intercepts of the K ‐factor model without making these restrictions. An almost sure approximation to that estimator is constructed here which is then used to obtain the asymptotic distribution of the GRS statistic. We call it the robust GRS statistic. Using data of Fama and French (2015, 2017), we use the robust GRS statistic to reconstruct their tables 5 and 4, respectively. As the distribution of the robust GRS does not change with the number of factors, in contrast to the finite sample version of this statistic, it allows for a more nuanced comparison of three‐, four‐ and five‐factor models. The power functions of the GRS (1989) statistic are compared with the enhanced version of the GRS appropriate for K factors.

In the statistical literature, ‘effect size’ is typically defined as the estimate of a fixed or random effects in a linear or other statistical model. Here, a more explicit measure of the estimate of effect size is studied, defined generically as X ¯ / S , where X ¯ and S are the sample mean and sample standard deviation for a random sample of observations X 1 , X 2 , …, X n from a distribution with finite mean μ , variance σ 2 and finite moments of order 3 and 4, so that skewness and kurtosis are well defined (see section 2 for explicit formulae for the skewness and kurtosis for a univariate real valued random variable). For example, in a paired comparison experiment, the X i would be the differences between treatment and control measurements. The effect size has also been referred to in signal processing as the ‘signal-to-noise-ratio,’ and presents a measure of how easy it is to detect a signal ( μ ) in the presence of noise quantified by σ. The larger the effect size, the higher is the confidence that a signal is present. Accordingly, it is of interest to find interval estimates of the population effect size μ /σ. In this note, the asymptotic (normal) distribution of n ( X ¯ / S − μ / σ ) is determined, and an explicit variance-stabilizing transformation g is determined such that n ( g ( X ¯ / S ) − g ( μ / σ ) ) is asymptotically standard normal, making confidence intervals for the effect size relatively easy to compute numerically. This explicit formula for a variance stabilizing transformation for the effect size appears to be a new result in the mathematical statistics literature, and there is strong evidence that confidence intervals formed after applying a variance-stabilizing transformation have merit over central-limit-theorem-based confidence intervals. Therefore, for a random sample from any completely known univariate distribution, this result allows one to compute a relatively efficient confidence interval for the effect size. AMS Subject Classification: 62-01

The heterochaos baker maps are piecewise affine maps on the square or the cube that are one of the simplest partially hyperbolic systems. The Dyck shift is a well-known example of a subshift that has two fully supported ergodic measures of maximal entropy (MMEs). We show that the two ergodic MMEs of the Dyck shift are represented as asymptotic distributions of sets of periodic points of different multipliers. We transfer this result to the heterochaos baker maps, and show that their two ergodic MMEs are represented as asymptotic distributions of sets of periodic points of different unstable dimensions.

Under the classical retrial policy, we consider a single-server M/G/1 queue with multiple input streams and orbits. Different types of customers have corresponding arrival rates, general distributions of service time and retrial rates. Assume that the retrial rates for different types of customers linearly converge to zero. We firstly derive the first-order asymptotics of the orbit queue lengths. Subsequently, we find that the joint asymptotic distribution of the number of retrials follows a multidimensional geometric distribution. Finally, we obtain the joint asymptotic distribution of waiting times, which follows a multidimensional exponential distribution. This result indicates that the waiting times for different types of customers are independent of each other.

ABSTRACT Detecting heterogeneity across latent subgroups is essential for understanding complex data structures in fields such as economics, medicine and social sciences. Threshold quantile regression (TQR) with change‐plane structures provides a flexible framework for such heterogeneity, especially in the presence of heavy‐tailed errors. However, a key challenge in subgroup detection lies in the nonidentifiability of threshold parameters under the null hypothesis, which invalidates classical testing approaches. To address this, we propose a difference of quantile loss test (DQLT) that constructs a supremum‐type test statistic based on differences in quantile loss functions. We derive the asymptotic distributions of the proposed test statistic under both the null and local alternative hypotheses and a bootstrap procedure is introduced to compute ‐values, effectively. The performance of our method is evaluated through extensive simulation studies, demonstrating well‐controlled type‐I error under the null hypothesis and enhanced power under the alternative hypotheses. We further apply the DQLT method to CHARLS 2015 and Boston housing datasets, illustrating its effectiveness in identifying subgroups with heterogeneous covariate effects.

The asymptotic uniform distribution of subset sums

Asymptotic distribution of the derivative of the taut string accompanying Wiener process

This research introduces a novel class of tests for assessing time series independence, utilizing Phi-entropy and a framework based on quantile symbolization. We derived the asymptotic distribution of the test statistic and developed a bootstrap version to ensure robust performance, which was subsequently validated for consistency. Simulations identified the optimal parameter values for the test, and comparisons with existing methods demonstrated its superior performance, particularly when using the Gini-Simpson index, which achieved the highest size-corrected power. Finally, applying the test to data from the Tehran Stock Exchange effectively detected dependencies, validated model fit, and confirmed the independence of residuals.

In the analysis of complex data from various fields like finance, imaging processing, and biomedical applications, the assumption regarding the structure of covariance holds a crucial role in ensuring accurate and efficient statistical inferences. We study the problem of evaluating whether a high-dimensional covariance matrix conforms to a linear structure defined by the linear combination of a predefined set of matrices. We introduce an innovative testing approach by integrating two Frobenius-type statistics, which capture both the difference and ratio between the unknown covariance matrix and the linearly structured matrix. Based on the joint distribution of the two test statistics, we derive the asymptotic null distribution and conduct a power analysis for this novel test under the high-dimensional setting. As evidenced by our extensive simulation studies, the proposed integrated test exhibits favorable control of the type I error rate under the null hypothesis, and demonstrates robust power across a spectrum of dense alternative hypotheses. Additionally, we further consider a power-enhanced test statistic that combines the proposed integrated test with a maximum-type test designed for sparse signals, enabling hypothesis testing against both dense and sparse alternatives.

Abstract We introduce a new type of test for complete spatial randomness that applies to mapped point patterns in a rectangle or a cube of any dimension. This is the first test of its kind to be based on characteristic functions and utilizes a weighted L2-distance between the empirical and uniform characteristic functions. The test shows surprising connections to Ripley’s K -function and Zimmerman’s ω2 statistic. It is also simple to calculate and does not require adjusting for edge effects. An efficient algorithm is developed to find the asymptotic null distribution of the test statistic under the Cauchy weight function. This makes the test fast to compute. In simulations, our test shows varying sensitivity to different levels of spatial interaction depending on the scale parameter of the Cauchy weight function. Tests with different parameter values can be combined to create a Bonferroni-corrected omnibus test, which is more powerful than the popular L -test and the Clark–Evans test in most simulation settings of heterogeneity, aggregation and regularity, especially when the sample size is large. The simplicity of the empirical characteristic function makes it straightforward to extend our test to non-rectangular or sparsely sampled point patterns.

Abstract The money exchange model is a type of agent-based model used to study how wealth distribution and inequality evolve through monetary exchanges between individuals. The primary focus of this model is to identify the limiting wealth distributions that emerge at the macroscopic level, given the microscopic rules governing the exchanges among agents. In this paper, we formulate generalized versions of the immediate exchange model, the uniform reshuffling model, and the uniform saving model, all of which are types of money exchange model, as discrete-time interacting particle systems and characterize their stationary distributions. Furthermore, we prove that, under appropriate scaling, the asymptotic wealth distribution converges to an exponential distribution for the uniform reshuffling model, and to either an exponential distribution or a gamma distribution depending on the tail behavior of the number of coins given/saved in the immediate exchange model and the random saving model, which generalizes the uniform saving model. In particular, our results provide a mathematically rigorous formulation and generalization of the assertions previously predicted in studies based on numerical simulations and heuristic arguments.

This paper offers an in-depth investigation into the Probability Weighted Moments (PWMs) methodology for estimating parameters of the Median Based Unit Weibull (MBUW) distribution. The author delves into a thorough comparison of the commonly employed first-order PWMs against more advanced higher-order PWMs. The analysis highlights the significant benefits associated with adopting these more sophisticated techniques, particularly in terms of accuracy and reliability in parameter estimation. In addition to this comparative analysis, the author derives the asymptotic distribution of the PWM estimator, which provides a theoretical foundation for the results and enhances the robustness of the conclusions. To further illustrate the practical implications of the findings, the author includes a detailed real data analysis that exemplifies the effectiveness of the proposed methodology. Through these examples, the author underscores the relevance of PWMs in real-world applications, demonstrating how this approach can lead to improved parameter estimates when working with the MBUW distribution.

An energy-stable full discretization for the modified elastic flow of closed curves is proposed. This is a gradient flow of a modified elastic energy combining bending and Dirichlet energies. The minimization of Dirichlet energy can lead to improved mesh quality. Gradient flows for both isotropic and anisotropic cases are considered. We derive new evolution equations for the parameterization and curvature vector of curves in arbitrary codimension. The proposed formulation is discretized by a parametric finite element method in space and a first-order implicit scheme in time. We establish the unconditional energy stability for the fully discretized scheme. Additionally, the second-order accuracy of the BDF2 scheme is demonstrated. Numerical examples in two and three dimensions illustrate the efficiency, energy stability, and asymptotic mesh distribution of the method for simulating the modified elastic flow.