Recently defined expectile regions capture the idea of centrality with respect to a multivariate distribution, but fail to describe the tail behavior while it is not at all clear what should be understood by a tail of a multivariate distribution. Therefore, cone expectile sets are introduced which take into account a vector preorder for the multi-dimensional data points. This provides a way of describing and clustering a multivariate distribution/data cloud with respect to an order relation. Fundamental properties of cone expectiles including dual representations of both expectile regions and cone expectile sets are established. It is shown that set-valued sublinear risk measures can be constructed from cone expectile sets in the same way as in the univariate case. Inverse functions of cone expectiles are defined which should be considered as rank functions rather than depth functions. Finally, expectile orders for random vectors are introduced and characterized via expectile rank functions.
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Spatially distributed functional data are prevalent in many statistical applications such as meteorology, energy forecasting, census data, disease mapping, and neurological studies. Given their complex and high-dimensional nature, functional data often require dimension reduction methods to extract meaningful information. Inverse regression is one such approach that has become very popular in the past two decades. We study the inverse regression in the framework of functional data observed at irregularly positioned spatial sites. The functional predictor is the sum of a spatially dependent functional effect and a spatially independent functional nugget effect, while the relation between the scalar response and the functional predictor is modeled using the inverse regression framework. For estimation, we consider local linear smoothing with a general weighting scheme, which includes as special cases the schemes under which equal weights are assigned to each observation or to each subject. This framework enables us to present the asymptotic results for different types of sampling plans over time such as non-dense, dense, and ultra-dense. We discuss the domain-expanding infill (DEI) framework for spatial asymptotics, which is a mix of the traditional expanding domain and infill frameworks. The DEI framework overcomes the limitations of traditional spatial asymptotics in the existing literature. Under this unified framework, we develop asymptotic theory and identify conditions that are necessary for the estimated eigen-directions to achieve optimal rates of convergence. Our asymptotic results include pointwise and $L_2$ convergence rates. Simulation studies using synthetic data and an application to a real-world dataset confirm the effectiveness of our methods.
Using dominating sets to separate vertices of graphs is a well-studied problem in the larger domain of identification problems. In such problems, the objective is typically to separate any two vertices of a graph by their unique neighbourhoods in a suitably chosen dominating set of the graph. Such a dominating and separating set is often referred to as a \emph{code} in the literature. Depending on the types of dominating and separating sets used, various problems arise under various names in the literature. In this paper, we introduce a new problem in the same realm of identification problems whereby the code, called the \emph{open-separating dominating code}, or the \emph{OSD-code} for short, is a dominating set and uses open neighbourhoods for separating vertices. The paper studies the fundamental properties concerning the existence, hardness and minimality of OSD-codes. Due to the emergence of a close and yet difficult to establish relation of the OSD-codes with another well-studied code in the literature called the open locating dominating codes, or OLD-codes for short, we compare the two on various graph classes. Finally, we also provide an equivalent reformulation of the problem of finding OSD-codes of a graph as a covering problem in a suitable hypergraph and discuss the polyhedra associated with OSD-codes, again in relation to OLD-codes of some graph classes already studied in this context.
For a sequence of random structures with $n$-element domains over a relational signature, we define its first order (FO) complexity as a certain subset in the Banach space $\ell^{\infty}/c_0$. The well-known FO zero-one law and FO convergence law correspond to FO complexities equal to $\{0,1\}$ and a subset of $\mathbb{R}$, respectively. We present a hierarchy of FO complexity classes, introduce a stochastic FO reduction that allows to transfer complexity results between different random structures, and deduce using this tool several new logical limit laws for binomial random structures. Finally, we introduce a conditional distribution on graphs, subject to a FO sentence $\varphi$, that generalises certain well-known random graph models, show instances of this distribution for every complexity class, and prove that the set of all $\varphi$ validating 0--1 law is not recursively enumerable.
In longitudinal observational studies with a time-to-event outcome, a common objective in causal analysis is to estimate the causal survival curve under hypothetical intervention scenarios within the study cohort. The g-formula is a particularly useful tool for this analysis. To enhance the traditional parametric g-formula approach, we developed a more adaptable Bayesian g-formula estimator. This estimator facilitates both longitudinal predictive and causal inference. It incorporates Bayesian additive regression trees in the modeling of the time-evolving generative components, aiming to mitigate bias due to model misspecification. Specifically, we introduce a more general class of g-formulas for discrete survival data. These formulas can incorporate the longitudinal balancing scores, which serve as an effective method for dimension reduction and are vital when dealing with an expanding array of time-varying confounders. The minimum sufficient formulation of these longitudinal balancing scores is linked to the nature of treatment regimes, whether static or dynamic. For each type of treatment regime, we provide posterior sampling algorithms, which are grounded in the Bayesian additive regression trees framework. We have conducted simulation studies to illustrate the empirical performance of our proposed Bayesian g-formula estimators, and to compare them with existing parametric estimators. We further demonstrate the practical utility of our methods in real-world scenarios using data from the Yale New Haven Health System's electronic health records.
Building prediction models from mass-spectrometry data is challenging due to the abundance of correlated features with varying degrees of zero-inflation, leading to a common interest in reducing the features to a concise predictor set with good predictive performance. In this study, we formally established and examined regularized regression approaches, designed to address zero-inflated and correlated predictors. In particular, we describe a novel two-stage regularized regression approach (ridge-garrote) explicitly modelling zero-inflated predictors using two component variables, comprising a ridge estimator in the first stage and subsequently applying a nonnegative garrote estimator in the second stage. We contrasted ridge-garrote with one-stage methods (ridge, lasso) and other two-stage regularized regression approaches (lasso-ridge, ridge-lasso) for zero-inflated predictors. We assessed the predictive performance and predictor selection properties of these methods in a comparative simulation study and a real-data case study to predict kidney function using peptidomic features derived from mass-spectrometry. In the simulation study, the predictive performance of all assessed approaches was comparable, yet the ridge-garrote approach consistently selected more parsimonious models compared to its competitors in most scenarios. While lasso-ridge achieved higher predictive accuracy than its competitors, it exhibited high variability in the number of selected predictors. Ridge-lasso exhibited slightly superior predictive accuracy than ridge-garrote but at the expense of selecting more noise predictors. Overall, ridge emerged as a favourable option when variable selection is not a primary concern, while ridge-garrote demonstrated notable practical utility in selecting a parsimonious set of predictors, with only minimal compromise in predictive accuracy.
Gene set analysis, a popular approach for analysing high-throughput gene expression data, aims to identify sets of genes that show enriched expression patterns between two conditions. In addition to the multitude of methods available for this task, users are typically left with many options when creating the required input and specifying the internal parameters of the chosen method. This flexibility can lead to uncertainty about the 'right' choice, further reinforced by a lack of evidence-based guidance. Especially when their statistical experience is scarce, this uncertainty might entice users to produce preferable results using a 'trial-and-error' approach. While it may seem unproblematic at first glance, this practice can be viewed as a form of 'cherry-picking' and cause an optimistic bias, rendering the results non-replicable on independent data. After this problem has attracted a lot of attention in the context of classical hypothesis testing, we now aim to raise awareness of such over-optimism in the different and more complex context of gene set analyses. We mimic a hypothetical researcher who systematically selects the analysis variants yielding their preferred results, thereby considering three distinct goals they might pursue. Using a selection of popular gene set analysis methods, we tweak the results in this way for two frequently used benchmark gene expression data sets. Our study indicates that the potential for over-optimism is particularly high for a group of methods frequently used despite being commonly criticised. We conclude by providing practical recommendations to counter over-optimism in research findings in gene set analysis and beyond.
We revisit the question of whether the strong law of large numbers (SLLN) holds uniformly in a rich family of distributions, culminating in a distribution-uniform generalization of the Marcinkiewicz-Zygmund SLLN. These results can be viewed as extensions of Chung's distribution-uniform SLLN to random variables with uniformly integrable $q^\text{th}$ absolute central moments for $0 < q < 2;\ q \neq 1$. Furthermore, we show that uniform integrability of the $q^\text{th}$ moment is both sufficient and necessary for the SLLN to hold uniformly at the Marcinkiewicz-Zygmund rate of $n^{1/q - 1}$. These proofs centrally rely on distribution-uniform analogues of some familiar almost sure convergence results including the Khintchine-Kolmogorov convergence theorem, Kolmogorov's three-series theorem, a stochastic generalization of Kronecker's lemma, and the Borel-Cantelli lemmas. The non-identically distributed case is also considered.
This paper develops a flexible and computationally efficient multivariate volatility model, which allows for dynamic conditional correlations and volatility spillover effects among financial assets. The new model has desirable properties such as identifiability and computational tractability for many assets. A sufficient condition of the strict stationarity is derived for the new process. Two quasi-maximum likelihood estimation methods are proposed for the new model with and without low-rank constraints on the coefficient matrices respectively, and the asymptotic properties for both estimators are established. Moreover, a Bayesian information criterion with selection consistency is developed for order selection, and the testing for volatility spillover effects is carefully discussed. The finite sample performance of the proposed methods is evaluated in simulation studies for small and moderate dimensions. The usefulness of the new model and its inference tools is illustrated by two empirical examples for 5 stock markets and 17 industry portfolios, respectively.
We propose a new numerical domain decomposition method for solving elliptic equations on compact Riemannian manifolds. One advantage of this method is its ability to bypass the need for global triangulations or grids on the manifolds. Additionally, it features a highly parallel iterative scheme. To verify its efficacy, we conduct numerical experiments on some $4$-dimensional manifolds without and with boundary.
Effect modification occurs when the impact of the treatment on an outcome varies based on the levels of other covariates known as effect modifiers. Modeling of these effect differences is important for etiological goals and for purposes of optimizing treatment. Structural nested mean models (SNMMs) are useful causal models for estimating the potentially heterogeneous effect of a time-varying exposure on the mean of an outcome in the presence of time-varying confounding. In longitudinal health studies, information on many demographic, behavioural, biological, and clinical covariates may be available, among which some might cause heterogeneous treatment effects. A data-driven approach for selecting the effect modifiers of an exposure may be necessary if these effect modifiers are \textit{a priori} unknown and need to be identified. Although variable selection techniques are available in the context of estimating conditional average treatment effects using marginal structural models, or in the context of estimating optimal dynamic treatment regimens, all of these methods consider an outcome measured at a single point in time. In the context of an SNMM for repeated outcomes, we propose a doubly robust penalized G-estimator for the causal effect of a time-varying exposure with a simultaneous selection of effect modifiers and prove the oracle property of our estimator. We conduct a simulation study to evaluate the performance of the proposed estimator in finite samples and for verification of its double-robustness property. Our work is motivated by a study of hemodiafiltration for treating patients with end-stage renal disease at the Centre Hospitalier de l'Universit\'e de Montr\'eal.
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