Traditional mediation analysis methods face challenges when dealing with a large number of mediators. In practice, these challenges can be compounded by outliers and the complex relationships introduced by confounders. To address these issues, we propose a quantile-based partially linear mediation analysis method that can handle high-dimensional mediators and introduce the fully-connected-deep-neural-network techniques to model intricate relationships in confounders. Unlike most existing works that focus on mediator selection, we emphasize inference on mediation effects. Theoretical analysis establishes asymptotic validity of the proposed tests, including control of type I error under the null and explicit power characterization under local alternatives. When the dimension of candidate mediators is high, the proposed method consistently selects important features in the outcome model. Numerical studies show that the proposed method outperforms existing approaches under a variety of settings, demonstrating its versatility and reliability as a modeling tool for complex data. We demonstrate the proposed method by examining the mediating role of DNA methylation in the relationship between childhood trauma and cortisol stress reactivity.
We investigate the problem of density estimation on the unit circle and the unit sphere from a computational perspective. Our primary goal is to develop new density estimators that are both rate-optimal and computationally efficient for direct implementation. After establishing these estimators, we derive closed-form expressions for probability estimates over regions of the circle and the sphere. Then, the proposed theories are supported by extensive simulation studies. The considered settings naturally arise when analyzing phenomena on the Earth's surface or in the sky (sphere), as well as directional or periodic phenomena (circle). The proposed approaches are broadly applicable, and we illustrate their usefulness through case studies in zoology, climatology, geophysics, and astronomy, which may be of independent interest. The methodologies developed here can be readily applied across a wide range of scientific domains.
Empirical likelihood is a popular nonparametric framework for inference due to its flexibility in model assumptions. The resulting profile empirical likelihood function shares similar properties with the parametric likelihood. According to Qin and Lawless [23], there exists a consistent local maximum within a small neighbourhood of the true parameter value. However, it is usually unknown whether a given local maximum is close to the true parameter value. To fill this important gap, we identify a set of conditions under which the global maximum of the profile empirical likelihood function is consistent. We subsequently propose a 'global consistency remedy' to ensure this property. We further propose a 'global maximum test' to ascertain whether a given local maximum is, in fact, the global maximum.
We present a solution to a question posed by G. R. Shorack and J. A. Wellner in Empirical Processes with Applications to Statistics [18]. Specifically, we derive the asymptotic distribution of their T-n(2) statistic. This statistic is related to Watson's statistic in the same way that Anderson-Darling's statistic is related to Cramer-von Mises' statistic. The authors conjectured that under uniformity, the asymptotic distribution of their statistic is an infinite weighted sum of independent chi-squared random variables. We prove this conjecture and compute the weights. The second part of the problem was related to the power of the test for uniformity based upon T-n(2), against specific alternatives. By proving that T-n(2) is a Beran's statistic, we show that Shorack-Wellner's test is a locally most powerful invariant test against certain circular alternatives. We prove the existence of these alternatives, outlining the shape of some of them. We derive a large-deviation type result for T-n(2). This result plays a key role in the theory of Bahadur efficiency. It enables us to compute the approximate and local exact slopes of the test for specific alternatives, including various classical parametric families. In all cases, a comparison with the Bahadur efficiency of Watson's statistic is provided.
The estimation problem of the common mean is studied when independent but heterogeneous observations have unknown variances. We discuss several invariant loss functions suitable for this purpose and determine the Bayes estimators against non-informative, objective priors. They have the form of a weighted average whose data-dependent weights are invariant under the location-scale transformations. In the case of normal data, conjugate prior for all parameters is shown to result in the Bayes estimator of the location parameter in t-distribution family. The asymptotic formulas for the variance of the considered estimators are provided.
