Cosmologies with positive λ: hierarchies of future behaviour.

Smooth Cauchy data on [Formula: see text] for the Einstein-[Formula: see text]-vacuum field equations with cosmological constant [Formula: see text] that are sufficiently close to de Sitter data develop into a solution that admits a smooth conformal boundary [Formula: see text] in its future. The conformal Einstein equations determine a smooth conformal extension across [Formula: see text] that defines on 'the other side' again a [Formula: see text]-vacuum solution. In this article, we discuss to what extent these properties generalize to the future asymptotic behaviour of solutions to the Einstein-[Formula: see text] equations with matter. We study Friedmann-Lemaitre-Robertson-Walker (FLRW) solutions and the Einstein-[Formula: see text] equations coupled to conformally covariant matter transport equations, to conformally privileged matter equations, and to conformally non-covariant matter equations. We present recent results on the Einstein-[Formula: see text]-perfect-fluid equations with a nonlinear asymptotic dust or asymptotic radiation equation of state. This article is part of a discussion meeting issue 'At the interface of asymptotics, conformal methods and analysis in general relativity'.

Distinguishing Between Long-Transient and Asymptotic States in a Biological Aggregation Model.

Aggregations are emergent features common to many biological systems. Mathematical models to understand their emergence are consequently widespread, with the aggregation-diffusion equation being a prime example. Here we study the aggregation-diffusion equation with linear diffusion in one spatial dimension. This equation is known to support solutions that involve both single and multiple aggregations. However, numerical evidence suggests that the latter, which we term 'multi-peaked solutions' may often be long-transient solutions rather than asymptotic steady states. We develop a novel technique for distinguishing between long transients and asymptotic steady states via an energy minimisation approach. The technique involves first approximating our study equation using a limiting process and a moment closure procedure. We then analyse local minimum energy states of this approximate system, hypothesising that these will correspond to asymptotic patterns in the aggregation-diffusion equation. Finally, we verify our hypotheses through numerical investigation, showing that our approximate analytic technique gives good predictions as to whether a state is asymptotic or transient. Overall, we find that almost all twin-peaked, and by extension multi-peaked, solutions are transient, except for some very special cases. We demonstrate numerically that these transients can be arbitrarily long-lived, depending on the parameters of the system.

Objective Priors for Invariant e-Values in the Presence of Nuisance Parameters.

This paper aims to contribute to refining the e-values for testing precise hypotheses, especially when dealing with nuisance parameters, leveraging the effectiveness of asymptotic expansions of the posterior. The proposed approach offers the advantage of bypassing the need for elicitation of priors and reference functions for the nuisance parameters and the multidimensional integration step. For this purpose, starting from a Laplace approximation, a posterior distribution for the parameter of interest is only considered and then a suitable objective matching prior is introduced, ensuring that the posterior mode aligns with an equivariant frequentist estimator. Consequently, both Highest Probability Density credible sets and the e-value remain invariant. Some targeted and challenging examples are discussed.

Debiased lasso for generalized linear models with a diverging number of covariates.

Modeling and drawing inference on the joint associations between single-nucleotide polymorphisms and a disease has sparked interest in genome-wide associations studies. In the motivating Boston Lung Cancer Survival Cohort (BLCSC) data, the presence of a large number of single nucleotide polymorphisms of interest, though smaller than the sample size, challenges inference on their joint associations with the disease outcome. In similar settings, we find that neither the debiased lasso approach (van de Geer et al., 2014), which assumes sparsity on the inverse information matrix, nor the standard maximum likelihood method can yield confidence intervals with satisfactory coverage probabilities for generalized linear models. Under this "large n, diverging p" scenario, we propose an alternative debiased lasso approach by directly inverting the Hessian matrix without imposing the matrix sparsity assumption, which further reduces bias compared to the original debiased lasso and ensures valid confidence intervals with nominal coverage probabilities. We establish the asymptotic distributions of any linear combinations of the parameter estimates, which lays the theoretical ground for drawing inference. Simulations show that the proposed refined debiased estimating method performs well in removing bias and yields honest confidence interval coverage. We use the proposed method to analyze the aforementioned BLCSC data, a large-scale hospital-based epidemiology cohort study investigating the joint effects of genetic variants on lung cancer risks.

Robust and efficient estimation of nonparametric generalized linear models.

Generalized linear models are flexible tools for the analysis of diverse datasets, but the classical formulation requires that the parametric component is correctly specified and the data contain no atypical observations. To address these shortcomings, we introduce and study a family of nonparametric full-rank and lower-rank spline estimators that result from the minimization of a penalized density power divergence. The proposed class of estimators is easily implementable, offers high protection against outlying observations and can be tuned for arbitrarily high efficiency in the case of clean data. We show that under weak assumptions, these estimators converge at a fast rate and illustrate their highly competitive performance on a simulation study and two real-data examples. Supplementary Information: The online version contains supplementary material available at 10.1007/s11749-023-00866-x.

Sequential adaptive strategies for sampling rare clustered populations.

A new class of sampling strategies is proposed that can be applied to population-based surveys targeting a rare trait that is unevenly spread over an area of interest. Our proposal is characterised by the ability to tailor the data collection to specific features and challenges of the survey at hand. It is based on integrating an adaptive component into a sequential selection, which aims both to intensify the detection of positive cases, upon exploiting the spatial clustering, and to provide a flexible framework to manage logistics and budget constraints. A class of estimators is also proposed to account for the selection bias, that are proved unbiased for the population mean (prevalence) as well as consistent and asymptotically Normal distributed. Unbiased variance estimation is also provided. A ready-to-implement weighting system is developed for estimation purposes. Two special strategies included in the proposed class are presented, that are based on the Poisson sampling and proved more efficient. The selection of primary sampling units is also illustrated for tuberculosis prevalence surveys, which are recommended in many countries and supported by the World Health Organisation as an emblematic example of the need for an improved sampling design. Simulation results are given in the tuberculosis application to illustrate the strengths and weaknesses of the proposed sequential adaptive sampling strategies with respect to traditional cross-sectional non-informative sampling as currently suggested by World Health Organisation guidelines.

Weak-instrument robust tests in two-sample summary-data Mendelian randomization.

Mendelian randomization (MR) has been a popular method in genetic epidemiology to estimate the effect of an exposure on an outcome using genetic variants as instrumental variables (IV), with two-sample summary-data MR being the most popular. Unfortunately, instruments in MR studies are often weakly associated with the exposure, which can bias effect estimates and inflate Type I errors. In this work, we propose test statistics that are robust under weak-instrument asymptotics by extending the Anderson-Rubin, Kleibergen, and the conditional likelihood ratio test in econometrics to two-sample summary-data MR. We also use the proposed Anderson-Rubin test to develop a point estimator and to detect invalid instruments. We conclude with a simulation and an empirical study and show that the proposed tests control size and have better power than existing methods with weak instruments.

Stability of asymptotic behaviour within polarized [Formula: see text]-symmetric vacuum solutions with cosmological constant.

We prove the nonlinear stability of the asymptotic behaviour of perturbations of subfamilies of Kasner solutions in the contracting time direction within the class of polarized [Formula: see text]-symmetric solutions of the vacuum Einstein equations with arbitrary cosmological constant [Formula: see text]. This stability result generalizes the results proven in Ames E et al. (2022 Stability of AVTD Behavior within the Polarized [Formula: see text]-symmetric vacuum spacetimes. Ann. Henri Poincaré. (doi:10.1007/s00023-021-01142-0)), which focus on the [Formula: see text] case, and as in that article, the proof relies on an areal time foliation and Fuchsian techniques. Even for [Formula: see text], the results established here apply to a wider class of perturbations of Kasner solutions within the family of polarized [Formula: see text]-symmetric vacuum solutions than those considered in Ames E et al. (2022 Stability of AVTD Behavior within the Polarized [Formula: see text]-symmetric vacuum spacetimes. Ann. Henri Poincaré. (doi:10.1007/s00023-021-01142-0)) and Fournodavlos G et al. (2020 Stable Big Bang formation for Einstein's equations: the complete sub-critical regime. Preprint. (http://arxiv.org/abs/2012.05888)). Our results establish that the areal time coordinate takes all values in [Formula: see text] for some [Formula: see text], for certain families of polarized [Formula: see text]-symmetric solutions with cosmological constant. This article is part of the theme issue 'The future of mathematical cosmology, Volume 1'.

LINEAR BIOMARKER COMBINATION FOR CONSTRAINED CLASSIFICATION.

Multiple biomarkers are often combined to improve disease diagnosis. The uniformly optimal combination, i.e., with respect to all reasonable performance metrics, unfortunately requires excessive distributional modeling, to which the estimation can be sensitive. An alternative strategy is rather to pursue local optimality with respect to a specific performance metric. Nevertheless, existing methods may not target clinical utility of the intended medical test, which usually needs to operate above a certain sensitivity or specificity level, or do not have their statistical properties well studied and understood. In this article, we develop and investigate a linear combination method to maximize the clinical utility empirically for such a constrained classification. The combination coefficient is shown to have cube root asymptotics. The convergence rate and limiting distribution of the predictive performance are subsequently established, exhibiting robustness of the method in comparison with others. An algorithm with sound statistical justification is devised for efficient and high-quality computation. Simulations corroborate the theoretical results, and demonstrate good statistical and computational performance. Illustration with a clinical study on aggressive prostate cancer detection is provided.

Long-range seed dispersal enables almost stationary patterns in a model for dryland vegetation.

Spatiotemporal patterns of vegetation are a ubiquitous feature of semi-arid ecosystems. On sloped terrain, vegetation patterns occur as stripes perpendicular to the contours. Field studies report contrasting long-term dynamics between different observation sites; some observe slow uphill migration of vegetation bands while some report stationary patterns. In this paper, we show that long-range seed dispersal provides a mechanism that enables the occurrence of both migrating and stationary patterns. We utilise a nonlocal PDE model in which seed dispersal is accounted for by a convolution term. The model represents vegetation patterns as periodic travelling waves and numerical continuation shows that both migrating and almost stationary patterns are stable if seed dispersal distances are sufficiently large. We use a perturbation theory approach to obtain analytical confirmation of the existence of almost stationary patterned solutions and provide a biological interpretation of the phenomenon.

Almost partition identities.

An almost partition identity is an identity for partition numbers that is true asymptotically [Formula: see text] of the time and fails infinitely often. We prove a kind of almost partition identity, namely that the number of parts in all self-conjugate partitions of n is almost always equal to the number of partitions of n in which no odd part is repeated and there is exactly one even part (possibly repeated). Not only does the identity fail infinitely often, but also, the error grows without bound. In addition, we prove several identities involving the number of parts in restricted partitions. We show that the difference in the number of parts in all self-conjugate partitions of n and the number of parts in all partitions of n into distinct odd parts equals the number of partitions of n in which no odd part is repeated, the smallest part is odd, and there is exactly one even part (possibly repeated). We provide both analytic and combinatorial proofs of this identity.

Rényi Entropies of Multidimensional Oscillator and Hydrogenic Systems with Applications to Highly Excited Rydberg States.

The various facets of the internal disorder of quantum systems can be described by means of the Rényi entropies of their single-particle probability density according to modern density functional theory and quantum information techniques. In this work, we first show the lower and upper bounds for the Rényi entropies of general and central-potential quantum systems, as well as the associated entropic uncertainty relations. Then, the Rényi entropies of multidimensional oscillator and hydrogenic-like systems are reviewed and explicitly determined for all bound stationary position and momentum states from first principles (i.e., in terms of the potential strength, the space dimensionality and the states's hyperquantum numbers). This is possible because the associated wavefunctions can be expressed by means of hypergeometric orthogonal polynomials. Emphasis is placed on the most extreme, non-trivial cases corresponding to the highly excited Rydberg states, where the Rényi entropies can be amazingly obtained in a simple, compact, and transparent form. Powerful asymptotic approaches of approximation theory have been used when the polynomial's degree or the weight-function parameter(s) of the Hermite, Laguerre, and Gegenbauer polynomials have large values. At present, these special states are being shown of increasing potential interest in quantum information and the associated quantum technologies, such as e.g., quantum key distribution, quantum computation, and quantum metrology.

Pandemic-type failures in multivariate Brownian risk models.

Modelling of multiple simultaneous failures in insurance, finance and other areas of applied probability is important especially from the point of view of pandemic-type events. A benchmark limiting model for the analysis of multiple failures is the classical d-dimensional Brownian risk model (Brm), see Delsing et al. (Methodol. Comput. Appl. Probab. 22(3), 927-948 2020). From both theoretical and practical point of view, of interest is the calculation of the probability of multiple simultaneous failures in a given time horizon. The main findings of this contribution concern the approximation of the probability that at least k out of d components of Brm fail simultaneously. We derive both sharp bounds and asymptotic approximations of the probability of interest for the finite and the infinite time horizon. Our results extend previous findings of DÈ©bicki et al. (J. Appl. Probab. 57(2), 597-612 2020) and DÈ©bicki et al. (Stoch. Proc. Appl. 128(12), 4171-4206 2018).

Reduced biomechanical models for precision-cut lung-slice stretching experiments.

Precision-cut lung-slices (PCLS), in which viable airways embedded within lung parenchyma are stretched or induced to contract, are a widely used ex vivo assay to investigate bronchoconstriction and, more recently, mechanical activation of pro-remodelling cytokines in asthmatic airways. We develop a nonlinear fibre-reinforced biomechanical model accounting for smooth muscle contraction and extracellular matrix strain-stiffening. Through numerical simulation, we describe the stresses and contractile responses of an airway within a PCLS of finite thickness, exposing the importance of smooth muscle contraction on the local stress state within the airway. We then consider two simplifying limits of the model (a membrane representation and an asymptotic reduction in the thin-PCLS-limit), that permit analytical progress. Comparison against numerical solution of the full problem shows that the asymptotic reduction successfully captures the key elements of the full model behaviour. The more tractable reduced model that we develop is suitable to be employed in investigations to elucidate the time-dependent feedback mechanisms linking airway mechanics and cytokine activation in asthma.

Directly and Simultaneously Expressing Absolute and Relative Treatment Effects in Medical Data Models and Applications.

Logistic regression is widely used in the analysis of medical data with binary outcomes to study treatment effects through (absolute) treatment effect parameters in the models. However, the indicative parameters of relative treatment effects are not introduced in logistic regression models, which can be a severe problem in efficiently modeling treatment effects and lead to the wrong conclusions with regard to treatment effects. This paper introduces a new enhanced logistic regression model that offers a new way of studying treatment effects by measuring the relative changes in the treatment effects and also incorporates the way in which logistic regression models the treatment effects. The new model, called the Absolute and Relative Treatment Effects (AbRelaTEs) model, is viewed as a generalization of logistic regression and an enhanced model with increased flexibility, interpretability, and applicability in real data applications than the logistic regression. The AbRelaTEs model is capable of modeling significant treatment effects via an absolute or relative or both ways. The new model can be easily implemented using statistical software, with the logistic regression model being treated as a special case. As a result, the classical logistic regression models can be replaced by the AbRelaTEs model to gain greater applicability and have a new benchmark model for more efficiently studying treatment effects in clinical trials, economic developments, and many applied areas. Moreover, the estimators of the coefficients are consistent and asymptotically normal under regularity conditions. In both simulation and real data applications, the model provides both significant and more meaningful results.

Identifying and characterising the impact of excitability in a mathematical model of tumour-immune interactions.

We study a five-compartment mathematical model originally proposed by Kuznetsov et al. (1994) to investigate the effect of nonlinear interactions between tumour and immune cells in the tumour microenvironment, whereby immune cells may induce tumour cell death, and tumour cells may inactivate immune cells. Exploiting a separation of timescales in the model, we use the method of matched asymptotics to derive a new two-dimensional, long-timescale, approximation of the full model, which differs from the quasi-steady-state approximation introduced by Kuznetsov et al. (1994), but is validated against numerical solutions of the full model. Through a phase-plane analysis, we show that our reduced model is excitable, a feature not traditionally associated with tumour-immune dynamics. Through a systematic parameter sensitivity analysis, we demonstrate that excitability generates complex bifurcating dynamics in the model. These are consistent with a variety of clinically observed phenomena, and suggest that excitability may underpin tumour-immune interactions. The model exhibits the three stages of immunoediting - elimination, equilibrium, and escape, via stable steady states with different tumour cell concentrations. Such heterogeneity in tumour cell numbers can stem from variability in initial conditions and/or model parameters that control the properties of the immune system and its response to the tumour. We identify different biophysical parameter targets that could be manipulated with immunotherapy in order to control tumour size, and we find that preferred strategies may differ between patients depending on the strength of their immune systems, as determined by patient-specific values of associated model parameters.

A fast score test for generalized mixture models.

In biomedical studies, testing for homogeneity between two groups, where one group is modeled by mixture models, is often of great interest. This paper considers the semiparametric exponential family mixture model proposed by Hong et al. (2017) and studies the score test for homogeneity under this model. The score test is nonregular in the sense that nuisance parameters disappear under the null hypothesis. To address this difficulty, we propose a modification of the score test, so that the resulting test enjoys the Wilks phenomenon. In finite samples, we show that with fixed nuisance parameters the score test is locally most powerful. In large samples, we establish the asymptotic power functions under two types of local alternative hypotheses. Our simulation studies illustrate that the proposed score test is powerful and computationally fast. We apply the proposed score test to an UK ovarian cancer DNA methylation data for identification of differentially methylated CpG sites.

A UNIFIED STUDY OF NONPARAMETRIC INFERENCE FOR MONOTONE FUNCTIONS.

The problem of nonparametric inference on a monotone function has been extensively studied in many particular cases. Estimators considered have often been of so-called Grenander type, being representable as the left derivative of the greatest convex minorant or least concave majorant of an estimator of a primitive function. In this paper, we provide general conditions for consistency and pointwise convergence in distribution of a class of generalized Grenander-type estimators of a monotone function. This broad class allows the minorization or majoratization operation to be performed on a data-dependent transformation of the domain, possibly yielding benefits in practice. Additionally, we provide simpler conditions and more concrete distributional theory in the important case that the primitive estimator and data-dependent transformation function are asymptotically linear. We use our general results in the context of various well-studied problems, and show that we readily recover classical results established separately in each case. More importantly, we show that our results allow us to tackle more challenging problems involving parameters for which the use of flexible learning strategies appears necessary. In particular, we study inference on monotone density and hazard functions using informatively right-censored data, extending the classical work on independent censoring, and on a covariate-marginalized conditional mean function, extending the classical work on monotone regression functions.

Counting and sampling gene family evolutionary histories in the duplication-loss and duplication-loss-transfer models.

Given a set of species whose evolution is represented by a species tree, a gene family is a group of genes having evolved from a single ancestral gene. A gene family evolves along the branches of a species tree through various mechanisms, including-but not limited to-speciation ([Formula: see text]), gene duplication ([Formula: see text]), gene loss ([Formula: see text]), and horizontal gene transfer ([Formula: see text]). The reconstruction of a gene tree representing the evolution of a gene family constrained by a species tree is an important problem in phylogenomics. However, unlike in the multispecies coalescent evolutionary model that considers only speciation and incomplete lineage sorting events, very little is known about the search space for gene family histories accounting for gene duplication, gene loss and horizontal gene transfer (the [Formula: see text]-model). In this work, we introduce the notion of evolutionary histories defined as a binary ordered rooted tree describing the evolution of a gene family, constrained by a species tree in the [Formula: see text]-model. We provide formal grammars describing the set of all evolutionary histories that are compatible with a given species tree, whether it is ranked or unranked. These grammars allow us, using either analytic combinatorics or dynamic programming, to efficiently compute the number of histories of a given size, and also to generate random histories of a given size under the uniform distribution. We apply these tools to obtain exact asymptotics for the number of gene family histories for two species trees, the rooted caterpillar and complete binary tree, as well as estimates of the range of the exponential growth factor of the number of histories for random species trees of size up to 25. Our results show that including horizontal gene transfers induce a dramatic increase of the number of evolutionary histories. We also show that, within ranked species trees, the number of evolutionary histories in the [Formula: see text]-model is almost independent of the species tree topology. These results establish firm foundations for the development of ensemble methods for the prediction of reconciliations.

Mixture distributions in a stochastic gene expression model with delayed feedback: a WKB approximation approach.

Noise in gene expression can be substantively affected by the presence of production delay. Here we consider a mathematical model with bursty production of protein, a one-step production delay (the passage of which activates the protein), and feedback in the frequency of bursts. We specifically focus on examining the steady-state behaviour of the model in the slow-activation (i.e. large-delay) regime. Using a formal asymptotic approach, we derive an autonomous ordinary differential equation for the inactive protein that applies in the slow-activation regime. If the differential equation is monostable, the steady-state distribution of the inactive (active) protein is approximated by a single Gaussian (Poisson) mode located at the globally stable fixed point of the differential equation. If the differential equation is bistable (due to cooperative positive feedback), the steady-state distribution of the inactive (active) protein is approximated by a mixture of Gaussian (Poisson) modes located at the stable fixed points; the weights of the modes are determined from a WKB approximation to the stationary distribution. The asymptotic results are compared to numerical solutions of the chemical master equation.