Monitoring both immunological and virological outcome measures provides the most accurate and meaningful assessment of the human immunodeficiency virus (HIV) treatment effectiveness. However, evaluating treatment efficacy based solely on either measure, using commonly employed survival and hazard functions, can introduce bias. In addition, heterogeneity in treatment-related recovery often results in skewed immunological and virological data. To address these challenges, we extend traditional survival analysis to the bivariate setting by introducing the bivariate median residual life function (MeRL) as a robust alternative for assessing treatment efficacy. In this framework, treatment efficacy is evaluated using the bivariate vector of time to immunological restoration and virological suppression. Unlike the mean residual life, the MeRL is less influenced by outliers and heavy-tailed distributions, which are frequently encountered in clinical studies. This paper proposes novel estimators for the bivariate MeRL function under order restrictions, and establishes their strong consistency and asymptotic distributions. Simulation studies were conducted to evaluate the performance of the proposed estimators. Finally, we demonstrated the practical utility of our approach through an application to real-world bivariate HIV data, offering valuable insights for both researchers and healthcarepractitioners.

The categorical Gini covariance is a measure of dependence between a numerical variable and a categorical variable, quantifying the difference between conditional and unconditional distribution functions. The categorical Gini covariance equals zero if and only if the numerical variable and the categorical variable are independent. Inspired by this property, we propose a non-parametric test to assess the independence between a numerical and categorical variable using a modified version of the categorical Gini covariance. We used the theory of U-statistics to find the test statistics and study the properties. The proposed test has an asymptotic normal distribution under both the null and alternative hypotheses. Since implementing a normal-based test is difficult, we developed a jackknife empirical likelihood (JEL) ratio test for testing independence. Monte Carlo simulation studies are performed to validate the performance of the proposed JEL ratio test. We illustrate the test procedure using two real data sets.

Instrumental variable (IV) methods are widely used to address unmeasured confoundings in structural equation models. In this paper, we focus on the settings where a possibly large number of instruments and a weak correlation between the instruments and the endogenous variable exist. Specifically, we propose a novel two-stage least squares (2SLS) model averaging approach to estimate the coefficient of an endogenous variable. Differing from existing literature, our model averaging estimation allows multiple exogenous variables to be included in both stages simultaneously. Theoretically, we study the consistency and asymptotic distributions of the estimated weights and the proposed model averaging estimator. Importantly, we discover that the proposed model averaging estimator produces an asymptotic bias when the endogenous variable and exogenous variables are correlated. Then, we construct a debiased estimator and establish its consistency and asymptotic normality to make statistical inference. Furthermore, we present an equivalent interpretation of the debiased estimator from another construction. Finally, numerical simulations and a real data analysis are conducted to illustrate our proposal.

Covariate-constrained randomization is an effective treatment allocation procedure for controlling imbalance across multiple baseline covariates in randomized trials. Motivated by the GroupPMPlus cluster randomized trial, we introduce the asymptotic theory for a broad class of estimators, known as M-estimators, under covariate-constrained randomization. Here, M-estimators refer to estimators obtained by optimizing an objective function, such as a log-likelihood function, and include commonly used methods such as analysis of covariance and linear mixed models. We show that M-estimators remain consistent in this setting but can exhibit non-Gaussian asymptotic distributions depending on the specification. Using examples of common M-estimators, we delineate conditions under which covariate-constrained randomization can be safely ignored in statistical analysis. Our results extend to stratified covariate-constrained randomization and semiparametric efficient estimators based on data-adaptive machine learning methods. We illustrate these theoretical findings using the GroupPMPlus study to evaluate the causal effect of a psychological treatment on mental health outcomes following a disaster.

Phylogenetic trees represent the evolutionary relationships between extant lineages, where extinct or non-sampled lineages are omitted. Extending the work of Stadler and collaborators, this paper focuses on the branch lengths in phylogenetic trees arising under a constant-rate birth-death model. We derive branch length distributions of phylogenetic branches with and without random sampling of individuals of the extant population under two distinct statistical scenarios: a fixed age of the birth-death process and a fixed number of individuals at the time of observation. We find that branches connected to the tree leaves (pendant branches) and branches in the interior of the tree behave very differently under sampling; pendant branches grow longer without limit as the sampling probability is decreased, whereas the interior branch lengths quickly reach an asymptotic distribution that does not depend on the sampling probability.

The Wilcoxon signed rank test statistic (T+) is one of the popular nonparametric test for one sample and paired sample data. As the sample size getting large, it is well known that this test statistic can be approximated by a normal distribution [1]. The Gibbons and Chakraborti assumes the independence between Zi and r(Di) and T+=sum(sum1<=i<=j<=N(Tij)) in their proof of asymptotic normality of Wilcoxon signed rank test statistic T+ , however these assumptions are not proved. In this work, we demonstrate the rigorous proof of both of these assumptions as two theorems to complete Gibbons and Chakraborti’s proof of asymptotic distribution of the Wilcoxon signed rank test statistics T+ and provide applications for these two theorems.

Current forecast evaluation tests can only assess forecasts generated at the same frequency. However, in real-world scenarios, predictions for the same economic variable may be made at different frequencies. The existing literature lacks statistical tests for evaluating such forecasts. This paper introduces a new evaluation test for comparing these forecasts. We propose a two-sample t-type test designed to test the equality of means between two potentially correlated time series that are sampled at different frequencies. No current estimator can compute the variance needed to studentize the difference in the sample means, given the potential temporal and cross-correlations, alongside the mixed-frequency nature of the related data. To address this, we propose a block-average-based variance estimator for this purpose. We derive the asymptotic null distribution of our new two-sample t statistic and analyze the test’s local power. Through extensive Monte Carlo simulations, our two-sample test demonstrates favorable size and power properties in finite samples. Notably, in cases with small sample sizes, our approach outperforms existing heuristic methods, which involve discarding data to align forecasts and applying conventional tests. Additionally, we uncover interesting connections with relevant methods in the literature.

Abstract This paper introduces a novel density estimator that combines an initial parametric approximation with a boundary-aware correction factor based on a semiparametric data transformation. The main contributions include achieving bias reduction comparable to existing semiparametric methods while simultaneously reducing estimation variance more effectively than current techniques, and developing a MISE-optimal plug-in bandwidth selector based on the initial parametric density estimator. The asymptotic distribution of the proposed data–driven bandwidth and its faster convergence to the ‘ideal’ bandwidth, compared to standard nonparametric methods, are established analytically herein. The improvement in finite sample estimation performance is demonstrated analytically as well as through both simulations and real data analysis, particularly in scenarios involving complex density features, such as multiple modes.

This article develops a simple and new approach for testing linear hypotheses about autocorrelations for time series with general stationary serial correlation structures. A practically important special case is the computation of robust confidence intervals for individual autocorrelations that do not require resampling methods. Inference is heteroscedasticity and autocorrelation robust (HAR) and allows innovations to be uncorrelated but not necessarily iid. It is well-known that the classic Bartlett formula can provide invalid inference when innovations are not independent and identically distributed (iid). While asymptotic variance formulas have been obtained under weaker assumptions, those formulas are complicated and resampling methods have been suggested in place of direct estimation of the variance. As an alternative we provide an easy to implement regression approach for estimating autocorrelations and their variance matrices. The asymptotic variance takes a sandwich form which can be estimated using well known HAR variance estimators. Resulting test statistics can be implemented with fixed-smoothing critical values which reduce finite sample size distortions. Monte Carlo simulations show our approach is robust to innovations that are not iid and works reasonably well across various serial correlation structures. An empirical illustration using robust confidence intervals for autocorrelations of S&P 500 index returns shows that conclusions about market efficiency and volatility clustering during pre and post-Covid periods using our approach contrast with conclusions using traditional (and often incorrectly used) methods. We provide a Python implementation for practitioners that implements our approach: https://eastlansing.github.io/Robust_CI_Acf/

Visual orthopedic gait assessment in dogs is recognized as subjective and is limited by interobserver variability. Objective detection of lameness is offered by biomechanical analysis, where asymmetry between limbs is quantified through kinematic parameters and symmetry indices. However, the minimum number of trials (full stride cycles) required to reliably discriminate lameness has remained a challenge. In this study, six healthy adult dogs were used. Mild, reversible lameness was induced in one forelimb using a cotton pad. Dogs were walked along a straight runway, and kinematic data were captured with a high-speed video camera. Stride length (SLE), support time (ST), and elbow range of motion (ROM) were measured. Symmetry indices (for linear and temporal parameters) and the symmetry angle (for angular parameters) were computed. The asymptotic distribution of these indices was derived using the delta method, which allowed for the construction of confidence intervals (CIs) and hypothesis tests for an asymmetry threshold of 3%. The number of trials required to achieve reliable detection was estimated through statistical simulations. Results indicated that the required number of trials was highly dependent on both the kinematic parameter and the magnitude of asymmetry. While detecting subtle asymmetries (≈4%) required a high number of trials (up to 347 for stride length), the requirements decreased substantially for more pronounced lameness. For a true asymmetry of 6%, 11-39 trials per limb were sufficient to achieve 80-90% power. It is concluded that the collection of only five trials is insufficient for detecting mild asymmetries. A statistical framework and practical recommendations for kinematic gait studies in dogs are provided.

Rooted binary perfect phylogenies provide a generalization of rooted binary unlabeled trees. In a rooted binary perfect phylogeny, each leaf is assigned a positive integer value that corresponds in a biological setting to the count of the number of indistinguishable lineages associated with the leaf. For the rooted binary unlabeled trees, these integers equal 1. We enumerate rooted binary perfect phylogenies with leaves and sample size , : the rooted binary unlabeled trees with leaves in which a sample of size lineages is distributed across the leaves. (1) First, we recursively enumerate rooted binary perfect phylogenies with sample size , summing over all possible , . We obtain an equation for the generating function, showing that asymptotically, the number of rooted binary perfect phylogenies with sample size grows with , faster than the rooted binary unlabeled trees, which grow with ≈ . (2) Next, we recursively enumerate rooted binary perfect phylogenies with a specific number of leaves and sample size . We report closed-form counts of the rooted binary perfect phylogenies with sample size and leaves. We provide a recurrence for the generating function describing, for each number of leaves , the number of rooted binary perfect phylogenies with leaves as the sample size increases. We also obtain an equation satisfied by the bivariate generating function counting rooted binary perfect phylogenies with leaves and sample size , as well as an asymptotic normal distribution for the number of leaves in a randomly chosen perfect phylogeny with sample size . (3) We find a generating function for the number of rooted binary perfect phylogenies with the -leaf caterpillar shape, growing with . We also find a generating function and exact count for the number of rooted binary perfect phylogenies with sample size and any caterpillar tree shape. A bivariate generating function counting rooted binary perfect phylogenies with leaves, sample size , and a caterpillar shape produces an asymptotic normal distribution for the number of leaves in a randomly chosen caterpillar perfect phylogeny with sample size . (4) Finally, we provide initial results recursively enumerating rooted binary perfect phylogenies with any specific unlabeled tree shape and sample size . The enumerations further characterize the rooted binary perfect phylogenies, which include the rooted binary unlabeled trees, and which can provide a set of structures useful for various biological contexts.

The density power divergence (DPD) is a well-studied member of the Bregman divergence family and forms the basis of widely used minimum divergence estimators that balance efficiency and robustness. In this paper, we introduce and study a new sub-class of Bregman divergences, termed the exponentially weighted divergence (EWD), designed to generate competitive and practically interpretable inference procedures. The EWD is constructed so that its associated weight function remains bounded within the interval [0, 1], which facilitates a transparent interpretation of robustness through controlled downweighting of low-density observations and avoids excessive influence from high-density points. We develop minimum EWD estimators (MEWDEs) within a general framework accommodating independent but non-homogeneous data, thereby extending classical minimum divergence theory beyond the i.i.d. setting. Under standard regularity conditions, we establish Fisher consistency and asymptotic normality, and we analyze robustness properties through influence function calculations. The EWD framework is further extended to parametric hypothesis testing, for which we derive the asymptotic null distribution of a Bregman divergence-based test statistic. Extensive simulation studies and real-data applications demonstrate that the proposed estimators perform comparably to, and often more robustly than, existing DPD-based procedures, particularly under moderate to heavy contamination, while retaining high efficiency under clean data. Overall, the EWD provides a tractable and interpretable alternative within the Bregman divergence class for robust parametric estimation and testing.

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.

Statistical modelling of bivariate data with ties is an active area of research. A widely used model for such data is the Marshall–Olkin bivariate exponential (MOBE) distribution. However, bivariate lifetime data often contain outliers, and the maximum likelihood estimator (MLE) for the MOBE model is highly sensitive to contamination. Therefore, robust parameter estimation is essential. In this article, we extend the minimum density power divergence estimation (MDPDE) method, proposed by Basu et al., to obtain robust and efficient estimation of the MOBE parameters. The MDPDE provides a robust generalization of the MLE while retaining good efficiency. We derive explicit estimating equations and establish the asymptotic distribution of the proposed estimators. The asymptotic relative efficiency is also investigated. The influence function of the MDPDE is bounded, ensuring robustness. A data–driven procedure for selecting the optimal tuning parameter is discussed. Simulation studies and a data application demonstrate the effectiveness of the proposed method overall.

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.

Abstract Lorenz dominance is a classical criterion for comparing income distributions with respect to inequality and social welfare. However, its binary nature, in which one distribution either dominates another or does not, often leads to inconclusive results when empirical Lorenz curves intersect. To overcome this limitation, we introduce the Lorenz dominance index (LDI), a continuous measure that quantifies the extent to which one Lorenz curve lies above another. The LDI provides an interpretable assessment based on the population, allowing for the evaluation of partial or near dominance and improving its usefulness in empirical settings. We derive the asymptotic distribution of the LDI and propose a nonparametric bootstrap procedure to construct confidence intervals and perform inference. Monte Carlo simulations confirm the estimator’s strong performance in finite samples and its nominal coverage. An application to household income data from China highlights the practical value of the LDI in distributional analysis.

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.

We propose new goodness-of-fit tests for exponentiality based on progressively Type-II censored data. These tests utilize scale-invariant statistics obtained from the Mahalanobis norm of normalized order statistics, leading to three test statistics, corresponding to L 2 -, L 1 -, and L ∞ -norms of centered uniform spacings. Exact and asymptotic distributions of these statistics are presented. A power study evaluates the proposed tests against existing benchmarks across various alternative distributions and censoring plans, demonstrating superior performance in cases with small and moderate sample sizes. Furthermore, we extend the methodology to approximate goodness-of-fit tests for Weibull distributions via power transformation, ensuring robustness w.r.t. the approximated significance level under unknown shape parameters. An illustrative data example confirms the practical applicability of our tests. Our findings highlight the potential for further extending goodness-of-fit tests under progressive Type-II censoring to other null distributions.

We derive the asymptotic distribution of ordinal-pattern frequencies under weak-dependence conditions and investigate the long-run covariance matrix not only analytically for moving-average, Gaussian, and the novel generalized coin-tossing processes, but also approximately by a simulation-based approach. Then, we deduce the asymptotic distribution of the entropy-complexity pair, which emerged as a popular tool for summarizing the time-series dynamics. Here, we make the necessary distinction between a uniform and a non-uniform ordinal pattern distribution and, thus, obtain two different limit theorems. On this basis, we consider a test for serial dependence and check its finite-sample performance. Moreover, we use our asymptotic results to approximate the estimation uncertainty of entropy-complexity pairs.