Abstract This paper is motivated by the problem of optimal allocation of trials in multi‐environment crop variety testing with a large number of varieties. Optimizing the allocation of trials results in the minimization of a design criterion with a Kronecker product structure in the information matrix. We consider the Kronecker–Bayesian linear criterion, which generalizes this design problem and has the form of the trace of the inverse of a sum of two Kronecker products. We derive a new general formula for the inverse of the sum of two Kronecker products, and we use this result to rewrite the Kronecker–Bayesian criterion in the form of the compound Bayes risk criterion, which can be recognized as a sum of Bayesian linear criteria with the same moment matrix. Based on the convexity and differentiability of the Kronecker–Bayesian linear criterion, we establish optimality conditions for approximate designs. We also propose a dimension reduction approach that provides highly efficient approximations for optimal designs. The proposed method allows for the preselection of an upper bound on the efficiency loss, which is independent of the true optimal design. Optimal or highly efficient designs can be computed under any kind of additional linear constraints, such as cost constraints. We apply our results to the problem of optimizing the allocation of trials in multi‐environment crop variety testing, and we illustrate the behavior of the optimal designs by real data examples. Finally, we consider further applications of the general formula for computing the inverse of the sum of two Kronecker products in control theory or multivariate time series analysis.

Abstract Many incomplete‐data statistical inference procedures are developed under the missing at random (MAR) assumption. However, the MAR assumption has been criticized as being overly strong for real‐data problems, and is unverifiable by using observed data. To handle data that are missing not at random (MNAR), sensitivity analysis has been proposed to investigate how conclusions are perturbed if the unverifiable MAR assumption is violated to a certain degree. This article proposes a new framework called multiple sensitivity models (MSMs) for performing general parameter estimation with the generalized estimating equation (GEE) method. Given user‐specified sensitivity parameters, a range of estimators is derived by solving the roots of the bounds of MSM‐assisted GEEs. Furthermore, we derive a representation for the proposed estimator so that it can be decomposed into several simpler estimators. It allows us to investigate the impact of different missing patterns. An asymptotically valid percentile bootstrap confidence region (CR) is also proposed. Theoretical justification is provided together with empirical evidence, which verifies the usefulness of the proposal's sensitivity analysis.

Abstract In this paper, we consider dependent competing risks using the Extended Marshall–Olkin model, where observation times are subject to censoring. The probabilistic properties of the survival copula associated with this model are examined, along with its application to the analysis of censored data. An estimation strategy using Bernstein polynomials for marginal distributions and joint survival probabilities is developed. The asymptotic normality of the proposed estimators is established under appropriate regularity conditions. The effectiveness of the proposed methodology is assessed through a simulation study using synthetic datasets and further validated with an application to real‐world data.

Abstract Harmonizable processes form a wide class of nonstationary processes, which admit a convenient Fourier analysis and have spectral distributions characterized by correlated components. They have been proven to be useful in many fields of application, e.g., in communication, seismology, EEG data analysis, etc. In this paper, we introduce a parametric form for harmonizable processes, namely Harmonizable Vector AutoRegressive and Moving Average models (HVARMA). In the same spirit as standard VARMA models, they are derived as a unique solution of a difference equation based on a properly defined concept of harmonizable noise. We exhibit their spectral characteristics and derive results for least‐squares parameter estimation in a fundamental case. We notably obtain, in some particular cases of the explosive regime, unusual asymptotic laws. Most importantly, we provide an effective way to generate realizations from those novel processes. Our modeling choice for harmonizable noise induces second‐order stationary dependencies among the components of the spectral distribution of the harmonizable time series. We choose to model them using a periodic stationary VARMA process, resulting in the so‐called HVARMA– model. We characterize its spectral properties and illustrate its ability to capture a vast range of nonstationary behaviors through examples of realizations using various models.

Abstract We propose a novel measure of statistical depth, the metric spatial depth, for data residing in an arbitrary metric space. The measure assigns high (low) values for points located near (far away from) the bulk of the data distribution, allowing quantifying their centrality/outlyingness. This depth measure is shown to have highly interpretable properties, making it appealing in object data analysis where standard descriptive statistics are difficult to compute. The proposed measure reduces to the classical spatial depth in a Euclidean space. In addition to studying its theoretical properties, to provide intuition on the concept, we explicitly compute metric spatial depths in several different metric spaces. Finally, we showcase the practical usefulness of the metric spatial depth in outlier detection, non‐convex depth region estimation and classification.

Abstract The Gaussian assumption is the most common and widely used distribution in statistical methodology. Various affine invariant tests for normality rely on the inverse of the covariance matrix and do not apply to high‐dimensional data. The Gaussian moments are equal to the specific values, which implies an exclusive quantitative relationship. This paper introduces two novel indices, the co‐third moment and the co‐fourth moment, to characterize the shape of the distribution relative to the high‐dimensional Gaussian family. Using these indices, two new tests with asymptotic properties under mild regularity conditions defining the subfamily for which tests are theoretically valid are proposed, which substantially avoid using the inverse covariance matrix and are easy to implement. By aggregating the strengths of the two tests and using the power enhancement technique, this study develops a more sensitive test of high‐dimensional normality. Numerical studies and real data analyses provide evidence that the proposed methods achieve well‐controlled size and competitive power, and illustrate their ability to detect departures from Gaussianity across different dimensional settings. The practical steps for empirically checking the regularity conditions are also discussed.

Abstract Hypothesis testing for fixed effects in linear mixed model is indispensable for investigating the utility of the predictors on response. However, when the dimension of covariates exceeds the sample size, the conventional frequentist methods designed for fixed dimensions fail completely. In this article, we develop a Bayesian‐motivated test for high‐dimensional linear mixed model to examine the significance of fixed effects in group. The proposed statistic is formulated as the ratio of two quadratic forms constructed from a sequence of independent but not identically distributed random variables. The null distribution of the proposed test statistic is derived through normality approximation for quadratic forms. To facilitate the implementation of the test, we introduce an innovative one‐step iteration method to determine the critical value. Additionally, the power function under local alternatives is derived under some mild conditions. In numerical experiments, we demonstrate the power performance in comparison with the existing method and the practical utility of the proposed method.

Abstract Inference in extreme value theory (EVT) relies on a limited number of extreme observations, making estimation challenging. To address this limitation, we propose a nonparametric simulation scheme, the multivariate extreme events spectral bootstrap simulation procedure, relying on the spectral representation of multivariate generalized Pareto‐distributed random vectors. Unlike standard bootstrap methods, our approach preserves the joint tail behavior of the data and generates additional synthetic extreme data, thereby improving the reliability of inference. We demonstrate the effectiveness of our procedure on the estimation of tail risk metrics, under both simulated and real data. The results highlight the potential of this method for enhancing risk assessment in high‐dimensional extreme scenarios.

Abstract Nonresponse in probability sampling presents a long‐standing challenge in survey sampling, often necessitating simultaneous adjustments to address sampling and selection biases. We develop a statistical framework that explicitly models sampling weights as random variables and establish the semiparametric efficiency bound for the parameter of interest under nonresponse. This study investigates strategies for eliminating bias and effectively utilizing available information, extending beyond nonresponse issues to data integration with external summary statistics. The proposed estimators are characterized by their efficiency and double robustness. However, realizing full efficiency hinges on the accurate specification of underlying models. To enhance robustness against potential model misspecification, we expand double robustness to multiple robustness through a novel two‐step empirical likelihood approach. A numerical study evaluates the finite‐sample performance of our methods. Additionally, we apply these methods to a dataset from the National Health and Nutrition Examination Survey, effectively integrating summary statistics from the National Health Interview Survey.

Abstract In high dimensions, the relationship between covariates and a response variable becomes increasingly intricate, with different covariate components often displaying varying degrees of variability. This complex interplay of dependence, heterogeneity, and high dimensionality presents a significant challenge when investigating the effects of covariates on the response variable. To address this, we propose a novel marginal testing procedure based on kernel‐based conditional mean dependence, which can be implemented without requiring model assumptions. Theoretically, we establish the limiting normal distributions of the test statistic under both null hypotheses and local alternatives by asymptotically approximating a class of quadratic forms. We also examine the asymptotic relative efficiency of the proposed test against several state‐of‐the‐art alternatives. Our theoretical evaluations are conducted from two perspectives: a detailed analysis within a linear model framework and a comparison within a fully non‐parametric setting. The effectiveness and applicability of the proposed method are demonstrated through both simulation studies and real data analysis.