Asymptotic Results Research Articles

Binary choice with misclassification and social interactions, with an application to peer effects in attitude

The interaction of economic agents is one of the most important elements in economic analyses. Social interactions on subjective outcomes, behavior, or decisions, are inherently difficult to identify and estimate because these variables are prone to misclassification errors. This paper puts forth a binary choice model with misclassification and social interactions to rectify the misclassification problems in social interactions studies. We achieve the identification of the conditional choice probability of the latent dependent variable by the technique of repeated measurements and a monotonicity condition. We construct the complete likelihood function from the two repeated measurements and propose a nested pseudo likelihood algorithm for estimation. Consistency and asymptotic normality results are shown for the proposed estimation method. We illustrate the finite sample performance of the model and the estimation method by three Monte Carlo experiments and an application to the study of peer effects among students in their attitudes towards learning.

Determination of the number of shots for Grover’s search algorithm

This paper focuses on Grover’s quantum search algorithm, which is of paramount importance as a masterpiece of Quantum Computing software. Given the inherent probabilistic nature of quantum computers, quantum programs based on Grover’s algorithm need to be run a number of times in order to generate a histogram of candidate values for solutions, which are then checked to identify the valid ones. In this paper, the distribution of the required number of shots to find all or a fraction of all the solutions to the Grover’s search problem is studied. Firstly, considering the similarity of the probability problem with the well-known coupon collector’s problem, two formulae are obtained from asymptotic results on the distribution of the required number of shots, as the number of problem solutions grows. These expressions allow to compute the number of shots required to ensure that, with probability p, all or a fraction of all the solutions are found. Secondly, the probability mass function of the required number of shots is derived, which serves as a benchmark to assess the validity of the asymptotic approximations derived previously. A comparison between the two approaches is presented and, as a result, a rule of thumb to decide under which circumstances employ one or the other is proposed.

Global asymptotic stability for Gurtin-MacCamy’s population dynamics model

In this paper, we investigate the global asymptotic stability of an age-structured population dynamics model with a Ricker’s type of birth function. This model is a hyperbolic partial differential equation with a nonlinear and nonlocal boundary condition. We prove a uniform persistence result for the semiflow generated by this model. We obtain the existence of global attractors and we prove the global asymptotic stability of the positive equilibrium by using a suitable Lyapunov functional. Furthermore, we prove that our global asymptotic stability result is sharp, in the sense that Hopf bifurcation may occur as close as we want from the region global stability in the space of parameter.

Asymptotics of lower dimensional zero-density regions

Topological data analysis (TDA) allows us to explore the topological features of a dataset. Among topological features, lower dimensional ones have recently drawn the attention of practitioners in mathematics and statistics due to their potential to aid the discovery of low dimensional structure in a data set. However, lower dimensional features are usually challenging to detect based on finite samples and using TDA methods that ignore the probabilistic mechanism that generates the data. In this paper, lower dimensional topological features occurring as zero-density regions of density functions are introduced and thoroughly investigated. Specifically, we consider sequences of coverings for the support of a density function in which the coverings are comprised of balls with shrinking radii. We show that, when these coverings satisfy certain sufficient conditions as the sample size goes to infinity, we can detect lower dimensional, zero-density regions with increasingly higher probability while guarding against false detection. We supplement the theoretical developments with the discussion of simulated experiments that elucidate the behaviour of the methodology for different choices of the tuning parameters that govern the construction of the covering sequences and characterize the asymptotic results.

Fast-slow wave transitions induced by a random mean flow.

Motivated by recent asymptotic results in atmosphere-ocean fluid dynamics, we present an idealized numerical and theoretical study of two-dimensional dispersive waves propagating through a small-amplitude random mean flow. The objective is to delineate clearly the conditions under which the cumulative Doppler shifting and refraction by the mean flow can change the group velocity of the waves not only in direction, but also in magnitude. The latter effect enables a possible transition from fast to slow waves, which behave very differently. Within our model we find the conditions on the dispersion relation and the mean flow amplitude that allow or rule out such fast-slow transitions. For steady mean flows we determine a finite mean flow amplitude threshold below which such transitions can be ruled out indefinitely. For unsteady mean flows a sufficiently rapid rate of change means that this threshold goes to zero, i.e., in this scenario all waves eventually undergo a fast-slow transition regardless of mean flow amplitude, with corresponding implications for the long-term fate of these waves.

SHORT-MATURITY ASYMPTOTICS FOR OPTION PRICES WITH INTEREST RATE EFFECTS

We derive the short-maturity asymptotics for option prices in the local volatility model in a new short-maturity limit [Formula: see text] at fixed [Formula: see text], where r is the interest rate and q is the dividend yield. In the case of practical relevance [Formula: see text] being small, however, our result holds for any fixed [Formula: see text]. The result is a generalization of the Berestycki–Busca–Florent formula ( Berestycki et al. , 2002 ) [Asymptotics and calibration of local volatility models, Quantitative Finance 2, 61–69] for the short-maturity asymptotics of the implied volatility which includes interest rates and dividend yield effects of [Formula: see text] to all orders in n. We obtain the analytical results for the ATM volatility and skew in this asymptotic limit. Explicit results are derived for the CEV model. The asymptotic result is tested numerically against the exact evaluation in the square-root model [Formula: see text], which demonstrates that the new asymptotic result is in very good agreement with the exact evaluation in a wide range of model parameters relevant for practical applications.

The asymptotic distribution of the permutation entropy.

Ordinal patterns serve as a robust symbolic transformation technique, enabling the unveiling of latent dynamics within time series data. This methodology involves constructing histograms of patterns, followed by the calculation of both entropy and statistical complexity-an avenue yet to be fully understood in terms of its statistical properties. While asymptotic results can be derived by assuming a multinomial distribution for histogram proportions, the challenge emerges from the non-independence present in the sequence of ordinal patterns. Consequently, the direct application of the multinomial assumption is questionable. This study focuses on the computation of the asymptotic distribution of permutation entropy, considering the inherent patterns' correlation structure. Furthermore, the research delves into a comparative analysis, pitting this distribution against the entropy derived from a multinomial law. We present simulation algorithms for sampling time series with prescribed histograms of patterns and transition probabilities between them. Through this analysis, we better understand the intricacies of ordinal patterns and their statistical attributes.

The dynamics of monkeypox disease under [formula omitted]–Hilfer fractional derivative: Application to real data

The mathematical model for monkeypox infection using the ψ–Hilfer fractional derivative is presented in this study. The integer order formulation is extended to the fractional order system by employing the ψ–Hilfer fractional derivative. The fractional order model analysis is provided. We investigate the model’s local asymptotical stability when R0<1. When R0>1, the global asymptotical stability result is displayed. We parameterize the model using recently reported cases of monkeypox infection in the United States. We calculated the basic reproduction using the estimated data and found it to be R0≈0.7121. We investigate the sensitivity of the monkeypox infection model and find the parameters that are sensitive R0. In general, we offer a numerical approach, and then for the monkeypox model, we present detailed findings. Some graphical outcomes for disease control in the United States are shown.

Linking within- and between-host scales for understanding the evolutionary dynamics of quantitative antimicrobial resistance

Understanding both the epidemiological and evolutionary dynamics of antimicrobial resistance is a major public health concern. In this paper, we propose a nested model, explicitly linking the within- and between-host scales, in which the level of resistance of the bacterial population is viewed as a continuous quantitative trait. The within-host dynamics is based on integro-differential equations structured by the resistance level, while the between-host scale is additionally structured by the time since infection. This model simultaneously captures the dynamics of the bacteria population, the evolutionary transient dynamics which lead to the emergence of resistance, and the epidemic dynamics of the host population. Moreover, we precisely analyze the model proposed by particularly performing the uniform persistence and global asymptotic results. Finally, we discuss the impact of the treatment rate of the host population in controlling both the epidemic outbreak and the average level of resistance, either if the within-host scale therapy is a success or failure. We also explore how transitions between infected populations (treated and untreated) can impact the average level of resistance, particularly in a scenario where the treatment is successful at the within-host scale.

The Cauchy–Dirichlet problem for the fast diffusion equation on bounded domains

The Fast Diffusion Equation (FDE) ut=Δum, with m∈(0,1), is an important model for singular nonlinear (density dependent) diffusive phenomena. Here, we focus on the Cauchy–Dirichlet problem posed on smooth bounded Euclidean domains. In addition to its physical relevance, there are many aspects that make this equation particularly interesting from the pure mathematical perspective. For instance: mass is lost and solutions may extinguish in finite time, merely integrable data can produce unbounded solutions, classical forms of Harnack inequalities (and other regularity estimates) fail to be true, etc.In this paper, we first provide a survey (enriched with an extensive bibliography) focussing on the more recent results about existence, uniqueness, boundedness and positivity (i.e., Harnack inequalities, both local and global), and higher regularity estimates (also up to the boundary and possibly up to the extinction time). We then prove new global (in space and time) Harnack estimates in the subcritical regime. In the last section, we devote a special attention to the asymptotic behaviour, from the first pioneering results to the latest sharp results, and we present some new asymptotic results in the subcritical case.

Couette-Poiseuille flow in a fluid overlying an anisotropic porous layer

In this study, Couette-Poiseuille flow inside a channel partially filled with a porous medium is treated. Considering the applicability to the practical scenarios, we assume that the porous medium is anisotropic in nature. More precisely, the model under steady-state assumption is described by the coupling of Brinkman-Forchheimer and Stokes equations. We develop the existence and uniqueness results corresponding to this two-dimensional Stokes-Brinkman-Forchheimer coupled system using the Browder-Minty theorem. For this we restrict the permeability to a diagonal matrix. The coupled system involves a non-linear term due to the Brinkman-Forchheimer equation. We propose an algorithm based on shooting and searching techniques to develop a numerical solution for this system, however, for an arbitrary anisotropic permeability matrix. The robustness of the presented numerical solution is established by comparing it with asymptotic solutions corresponding to small and large Darcy numbers. For the case of a large Darcy number, the problem of interest is a regular perturbation while the same is a singular perturbation problem for the case of a small Darcy number, which is dealt with using Prandtl's matching principle. With the velocity distribution determined, we have presented a comparative analysis of the shear stress distribution at the liquid-porous interface and at the bottom plate, which is of interest to analyze the stress distribution on the arterial wall. The asymptotic results are found to be in good agreement with those obtained numerically.

The Cauchy problem of the Camassa-Holm equation in a weighted Sobolev space: Long-time and Painlevé asymptotics

Based on the ∂‾-generalization of the Deift-Zhou steepest descent method, we extend the long-time and Painlevé asymptotics for the Camassa-Holm (CH) equation to the solutions with initial data in a weighted Sobolev space H4,2(R). With a new scale (y,t) and a Riemann-Hilbert problem associated with the initial value problem, we derive different long time asymptotic expansions for the solutions of the CH equation in different space-time solitonic regions. The half-plane {(y,t):−∞<y<∞,t>0} is divided into four asymptotic regions: 1. Fast decay region, y/t∈(−∞,−1/4) with an error O(t−1/2); 2. Modulation-soliton region, y/t∈(2,+∞), the result can be characterized with an modulation-solitons with residual error O(t−1/2); 3. Zakhrov-Manakov region, y/t∈(0,2) and y/t∈(−1/4,0). The asymptotic approximation is characterized by the dispersion term with residual error O(t−3/4); 4. Two transition regions, |y/t|≈2 and |y/t|≈−1/4, the asymptotic results are described by the solution of Painlevé II equation with error order O(t−1/2).

A perturbation analysis of pure torsion of incompressible hyperelastic cylinders

The stretch-based formulation of basic problems for incompressible isotropic hyperelastic materials has been the subject of renewed recent attention largely motivated by application to modelling the mechanical response of soft tissues. Here we are concerned with the classical problem of simple torsion of a circular cylinder which has in the past been primarily investigated by employing the classical approach of Rivlin in terms of invariants of the Cauchy–Green deformation tensors. Here we propose a perturbation approach to the pure torsion problem for thin (but not necessarily infinitesimally thin) cylindrical specimens motivated by potential applications to rubber-like or biological filaments and for the infinitesimal twisting of thick cylinders. The advantage of this approach is that the results are valid for all strain–energy functions whether based in terms of the principal stretches or in terms of the classical deformation invariants. The asymptotic results obtained for the resultant force and moment are applied to the special case of the one-term stretch-based Ogden model. In particular, a prediction of a transition in the Poynting effect for values of the Ogden parameter α>6 is obtained. Other illustrative examples for both invariant and stretch-based strain energies are described. In view of the complexity of a general analysis of torsion for arbitrary strain energies, the perturbation approach provides a viable alternative for assessing the resultant force and moment.

Asymptotic Results of Some Conditional Nonparametric Functional Parameters in High-Dimensional Associated Data

In this paper, we propose to study the asymptotic properties of some conditional functional parameters, such as the distribution function, the density, and the hazard function, for an explanatory variable with values in a Hilbert space (infinite dimension) and a response variable real in a quasi-associated dependency framework. We consider the non parametric estimation of the conditional distribution function by the kernel method in the presence of the quasi-associated dependence, and we establish under general hypotheses the almost complete convergence with speed of the estimator built in the associated case. The estimation of the conditional hazard function will be conducted by utilizing the two outcomes of the conditional distribution function and the conditional density. We establish the asymptotic normality of the kernel estimator as the conditional risk function of a properly normalized functional. We explicitly give the asymptotic variance. Simulation studies were conducted to investigate the behavior of the asymptotic property in the context of finite sample data. All the statistical analyses were performed using R software.

Machine‐learning‐based asymptotic homogenisation and localisation considering boundary layer effects

AbstractAsymptotic homogenisation offers a way to efficiently analyse the mechanical behaviour of multiscale configurations. But near a multiscale boundary, the homogenisation strategy should be modified, as the underlying periodicity assumption breaks down there. In this article, we introduce a machine‐learning‐based asymptotic homogenisation and localisation scheme to formulate such boundary layer effects. To this end, we define a set of boundary layer cells, where external loading conditions are imposed on one side of the cell, and matching conditions with the interior periodic cells are imposed on the opposite side. The formulation is also extended to cover situations where the multi‐scale structure is not fully periodic, but spatially varying. Implied from the asymptotic results, neural networks can be trained to memorise the interrelationship between key local quantities, such as the magnitude of the local maximum von Mises stress, and the local mechanical and geometric features. Equipped with the trained neural networks, the online calculation for key (boundary‐localised) quantities of interest under arbitrary loading conditions is expected to be accelerated substantially. Numerical examples are further presented to show the reliability of the proposed work for boundary stress prediction.

Arched beams of Bresse type: New thermal couplings and pattern of stability

This is the second paper of a trilogy intended by the authors in what concerns a unified approach to the stability of thermoelastic arched beams of Bresse type under Fourier’s law. Differently of the first one, where the thermal couplings are regarded on the axial and bending displacements, here the thermal couplings are taken over the shear and bending forces. Such thermal effects still result in a new prototype of partially damped Bresse system whose stability results demand a proper approach. Combining a novel path of local estimates by means of the resolvent equation along with a control-observability analysis developed for elastic non-homogeneous systems of Bresse type proposed in trilogy’s first paper, we are able to provide a unified methodology of the asymptotic stability results, by proving the pattern of them with respect to boundary conditions and the action of temperature couplings, which is in compliance with our previous and present goal.

Unimodal Maps Perturbed by Heteroscedastic Noise: An Application to a Financial Systems

We investigate and prove the mathematical properties of a general class of one-dimensional unimodal smooth maps perturbed with a heteroscedastic noise. Specifically, we investigate the stability of the associated Markov chain, show the weak convergence of the unique stationary measure to the invariant measure of the map, and show that the average Lyapunov exponent depends continuously on the Markov chain parameters. Representing the Markov chain in terms of random transformation enables us to state and prove the Central Limit Theorem, the large deviation principle, and the Berry-Esséen inequality. We perform a multifractal analysis for the invariant and the stationary measures, and we prove Gumbel’s law for the Markov chain with an extreme index equal to 1. In addition, we present an example linked to the financial concept of systemic risk and leverage cycle, and we use the model to investigate the finite sample properties of our asymptotic results

Estimating the number of common trends in large [formula omitted] and [formula omitted] factor models via canonical correlations analysis

Asymptotic results for canonical correlations are derived when the analysis is performed between levels and cumulated levels of N time series of length T, generated by a factor model with s common stochastic trends. For T→∞ and fixed N and s, the largest s squared canonical correlations are shown to converge to a non-degenerate limit distribution while the remaining N−s converge in probability to 0. Furthermore, if s grows at most linearly in N, the largest s squared canonical correlations are shown to converge in probability to 1 as (T,N)seq→∞. This feature allows one to estimate the number of common trends as the integer with largest decrease in adjacent squared canonical correlations. The maximal gap equals 1 in the limit and this criterion is shown to be consistent. A Monte Carlo simulation study illustrates the findings.

On strongly dependent zero-inflated INAR(1) processes

We consider INAR(1) processes modulated by an unobserved strongly dependent 0-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0-1$$\\end{document} process. The observed process exhibits zero inflation and long memory. A simple method is proposed for estimating the INAR-parameters without modelling the unobserved modulating process. Asymptotic results for the estimators are derived, and a zero-inflation test is introduced. Asymptotic rejection regions and asymptotic power under long-memory alternatives are derived. A small simulation study illustrates the asymptotic results.

Robust analyzes for longitudinal clinical trials with missing and non-normal continuous outcomes

Missing data are unavoidable in longitudinal clinical trials, and outcomes are not always normally distributed. In the presence of outliers or heavy-tailed distributions, the conventional multiple imputation with the mixed model with repeated measures analysis of the average treatment effect (ATE) based on the multivariate normal assumption may produce bias and power loss. Control-based imputation (CBI) is an approach for evaluating the treatment effect under the assumption that participants in both the test and control groups with missing outcome data have a similar outcome profile as those with an identical history in the control group. We develop a robust framework to handle non-normal outcomes under CBI without imposing any parametric modeling assumptions. Under the proposed framework, sequential weighted robust regressions are applied to protect the constructed imputation model against non-normality in the covariates and the response variables. Accompanied by the subsequent mean imputation and robust model analysis, the resulting ATE estimator has good theoretical properties in terms of consistency and asymptotic normality. Moreover, our proposed method guarantees the analysis model robustness of the ATE estimation in the sense that its asymptotic results remain intact even when the analysis model is misspecified. The superiority of the proposed robust method is demonstrated by comprehensive simulation studies and an AIDS clinical trial data application.