Asymptotic Results Research Articles

Adaptive lasso and Dantzig selector for spatial point processes intensity estimation

Lasso and Dantzig selector are standard procedures able to perform variable selection and estimation simultaneously. This paper is concerned with extending these procedures to spatial point process intensity estimation. We propose adaptive versions of these procedures, develop efficient computational methodologies and derive asymptotic results for a large class of spatial point processes under an original setting where the number of parameters, i.e. the number of spatial covariates considered, increases with the expected number of data points. Both procedures are compared theoretically, in a simulation study, and in a real data example.

Nonlinear electrophoretic velocity of a spherical colloidal particle

Electrophoresis is the motion of a charged colloidal particle in an electrolyte under an applied electric field. The electrophoretic velocity of a spherical particle depends on the dimensionless electric field strength $\beta =a^*e^*E_\infty ^*/k_B^*T^*$ , defined as the ratio of the product of the applied electric field magnitude $E_\infty ^*$ and particle radius $a^*$ , to the thermal voltage $k_B^*T^*/e^*$ , where $k_B^*$ is Boltzmann's constant, $T^*$ is the absolute temperature, and $e^*$ is the charge on a proton. In this paper, we develop a spectral element algorithm to compute the electrophoretic velocity of a spherical, rigid, dielectric particle, of fixed dimensionless surface charge density $\sigma$ over a wide range of $\beta$ . Here, $\sigma =(e^*a^*/\epsilon ^*k_B^*T^*)\sigma ^*$ , where $\sigma ^*$ is the dimensional surface charge density, and $\epsilon ^*$ is the permittivity of the electrolyte. For moderately charged particles ( $\sigma ={O}(1)$ ), the electrophoretic velocity is linear in $\beta$ when $\beta \ll 1$ , and its dependence on the ratio of the Debye length ( $1/\kappa ^*$ ) to particle radius (denoted by $\delta =1/(\kappa ^*a^*)$ ) agrees with Henry's formula. As $\beta$ increases, the nonlinear contribution to the electrophoretic velocity becomes prominent, and the onset of this behaviour is $\delta$ -dependent. For $\beta \gg 1$ , the electrophoretic velocity again becomes linear in field strength, approaching the Hückel limit of electrophoresis in a dielectric medium, for all $\delta$ . For highly charged particles ( $\sigma \gg 1$ ) in the thin-Debye-layer limit ( $\delta \ll 1$ ), our computations are in good agreement with recent experimental and asymptotic results.

A calibration method to stabilize estimation with missing data

AbstractThe augmented inverse weighting (AIW) estimator is commonly used to estimate the marginal mean of an outcome because of its doubly robust property. However, the AIW estimator can be severely biased if both the propensity score (PS) and the outcome regression (OR) models are misspecified. One possible reason is that misspecification of the PS or OR model yields extreme values in these models, which can have a great influence on the marginal mean estimate. In this article, we propose a calibrated AIW estimator for the marginal mean, which can control the influence of these extreme values and provide a stable marginal mean estimator. The proposed estimator also enjoys the doubly robust property. We also extend this method to handle high‐dimensional covariates in PS and OR models. Asymptotic results are also developed. Extensive simulation studies show that the proposed method performs better in most cases than existing approaches by providing a more stable estimate. We apply this method to an AIDS clinical trial study.

Asymptotic formulations of anti-plane problems in pre-stressed compressible elastic laminates

Abstract This article investigates the long-wave anti-plane shear motion in a symmetric three-layered laminate composed of pre-stressed compressible elastic layers. The layers of the laminate are perfectly bonded, while traction-free and fixed boundary conditions are considered on the outer faces of the laminate. In both cases, the dispersion relation is obtained in terms of symmetric and anti-symmetric decompositions. Numerical results and an asymptotic long-wave analysis are presented, corresponding to the three possible vibration modes. It is revealed that a low-frequency mode only exists in respect of symmetric motion with free-faces, while all other cases pose a series of non-zero cut-off frequencies. Comparisons between the exact and approximate asymptotic results are presented, and excellent agreement is observed.

Finite‐time and fixed‐time adaptive consensus of multi‐agent systems with general linear dynamics

This paper investigates the adaptive consensus problem of general linear multi‐agent systems. Finite‐time consensus and fixed‐time consensus are both discussed. In order to remove the dependence on global information about system topology, distributed adaptive mechanisms are introduced. Compared with the existing asymptotic consensus results, the system's convergence speed is improved by the proposed distributed finite‐time consensus protocol. Furthermore, a distributed fixed‐time consensus protocol is proposed to remove the dependence on system's initial state. Under the fixed‐time consensus protocol, it is proved that an upper bound of the system's convergence time can be estimated. Sufficient conditions are given to ensure that all agents can achieve consensus in finite time and fixed time, respectively. Finally, the effectiveness of the proposed protocols is illustrated by a simulation example.

A Transmuted Modified Power-Generated Family of Distributions with Practice on Submodels in Insurance and Reliability

In this article, we propose a new transmuted modified power-generalized family of distributions constructed from the transmuted-generated and modified power-generated families. The proposed approach is flexible and provides a tradeoff between the two baseline families. For a prime study, we identify the main characteristics of the new transmuted modified power family, such as the asymptotic results, quantile function, series representation, and the various kinds of moment measures. By using the exponential distribution as the baseline, a new three-parameter lifetime distribution is constructed. The associated probability functions (density and hazard rate) are flexible and have a variety of asymmetric shapes, which make them attractive for statistical purposes. In particular, for the related probability density function, reversed-J, unimodal, and right-skewed shapes are observed. Measures relating to risk theory are also computed, such as the value at risk and the expected shortfall. By using both simulation analysis and the maximum likelihood approach, the estimation of the model parameters is evaluated. The effectiveness of the proposed model is demonstrated by two real-world cases (one in insurance and the other in reliability), and we show that it yields better fits when compared to other extended models connected to the exponential model.

Testing of Constant Parameters for Semi‐Parametric Functional Coefficient Models with Integrated Covariates

Cointegration has been widely used in macroeconomics and financial time series analysis, but traditional linear cointegration relationship is often rejected in empirical applications. Many constant parameters testing methods in semi‐parametric functional coefficient cointegrated framework have been developed accordingly. However, there are little studies on constant parameters testing problem for the case that the index variable is integrated of order one. From a practical point of view, there is also a need for a test that accommodates integrated index variable in functional coefficient cointegrated setting, for example, in the study of the purchasing power parity hypothesis. In this article, an orthogonal series approximation‐based test statistic is proposed to tackle the problem. The asymptotic results are also studied. Monte Carlo experiments are conducted to evaluate the finite sample performance of the proposed test, and an empirical example about price and exchange rate data is provided.

Asymptotics for sum-ruin probabilities of a bidimensional risk model with heavy-tailed claims and stochastic returns

This paper considers a bidimensional risk model with geometric Lévy price processes and dependent heavy-tailed claims, where the two claim-number processes generated by the two different lines of business are almost arbitrarily dependent. When the distributions of the claims are subexponential with a positive lower Matuszewska index, an asymptotic formula for the finite-time sum-ruin probability is derived, which has a more transparent form when the distributions of the claims are regularly-varying-tailed and the two claim-number processes are homogeneous Poisson processes. Some simulation studies are conducted to verify the accuracy and sensitivity of the asymptotic result by employing the crude Monte Carlo method.

Detailed asymptotic expansions for partitions into powers

In this paper, we examine the number of ways to partition an integer [Formula: see text] into [Formula: see text]th powers when [Formula: see text] is large. Simplified proofs of some asymptotic results of Wright are given using the saddle-point method, including exact formulas for the expansion coefficients. The convexity and log-concavity of these partitions is shown for large [Formula: see text], and the stronger conjectures of Ulas are proved. The asymptotics of Wright’s generalized Bessel functions are also treated.

Large deviations of the interface height in the Golubović-Bruinsma model of stochastic growth.

We study large deviations of the one-point height H of a stochastic interface, governed by the Golubović-Bruinsma equation, ∂_{t}h=-ν∂_{x}^{4}h+(λ/2)(∂_{x}h)^{2}+sqrt[D]ξ(x,t), where h(x,t) is the interface height at point x and time t and ξ(x,t) is the Gaussian white noise. The interface is initially flat, and H is defined by the relation h(x=0,t=T)=H. We focus on the short-time limit, T≪T_{NL}, where T_{NL}=ν^{5/7}(Dλ^{2})^{-4/7} is the characteristic nonlinear time of the system. In this limit typical, small fluctuations of H are unaffected by the nonlinear term, and they are Gaussian. However, the large-deviation tails of the probability distribution P(H,T) "feel" the nonlinearity already at short times, and they are non-Gaussian and asymmetric. We evaluate these tails using the optimal fluctuation method (OFM). The lower tail scales as -lnP(H,T)∼H^{5/2}/T^{1/2}. It coincides with its analog for the Kardar-Parisi-Zhang (KPZ) equation, and we point out to the mechanism of this universality. The upper tail scales as -lnP(H,T)∼H^{11/6}/T^{5/6}, it is different from the upper tail of the KPZ equation. We also compute the large deviation function of H numerically and verify our asymptotic results for the tails.

Neuroadaptive Asymptotic Tracking Control of Nonlinear Systems with Multiple Uncertainties

In this work, several innovative robust neuroadaptive control algorithms are integrated via the command filtered backstepping approach for a class of mismatched uncertain nonlinear systems. They utilize neural network adaptive control to achieve an enhanced model compensation effect. In addition, different robust terms with a positive time-varying integral expression are synthesized for each of the algorithms to address external disturbances. Each of the synthesized control algorithms has a continuous control input and cannot only remove the adverse effects of the “explosion of complexity” inherent in the traditional backstepping technology but also produce an asymptotic tracking result for the closed-loop system via strict theoretical analysis. Comparative simulation verifications are implemented to check the feasibility and practicability of the presented controllers.

An Asymptotic Result on Catastrophe Insurance Losses

Consider an insurer who both sells catastrophe insurance policies and makes risky investments. Suppose that insurance claims arrive according to a Poisson process and the price of the investment portfolio evolves according to a general stochastic process independent of the insurance claims. In the focus of catastrophe risk management are catastrophe insurance losses. For the case of heavy-tailed claims, we derive a simple asymptotic formula for the tail probability of the present value of future claims. The transparent expression of our formula explicitly reflects the different roles of the various underlying risks in driving catastrophic losses. Our work is distinguished from most other works in this strand of research in that we carry out the asymptotic study over the whole class of subexponential distributions. Thus, our work allows both very heavy-tailed distributions such as Pareto-type distributions and moderately heavy-tailed distributions such as Lognormal and Weibull distributions.

On the Study of Multiuser Mixed Dual-Hop RF/THz Systems

A multiuser mixed radio frequency/terahertz (MRFTHz) system is studied, where the multiple RF channels experience <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\eta$</tex-math></inline-formula> - <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mu$</tex-math></inline-formula> fading and the THz channel obeys <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha$</tex-math></inline-formula> - <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mu _{1}$</tex-math></inline-formula> fading with pointing errors (PEs). Both fixed-gain (FG) amplify-and-forward (AF) and decode-and-forward (DF) protocols are considered in the relay. Firstly, the exact expressions of the cumulative distribution functions of the overall signal-to-noise ratio under AF and DF schemes are derived. On this basis, the Fox's H-function (FHF) and bivariate FHF are used to obtain exact analytical expressions of the outage probability (OP), average bit error rate, and average channel capacity. Additionally, the asymptotic results at high signal-to-noise ratios (SNRs) are provided to gain the system's diversity order (DO). Results show that the DOs of the AF and DF relaying systems are decided by the PEs and channel fading parameters (FPs). In addition, considering the FG AF relaying and multi-antenna scenarios, the asymptotic OP at high SNRs is derived, which obtaining the DO is determined by the numbers of the antenna, PEs, and FPs. Finally, we apply the Monte Carlo simulations to verify numerical results. Results show that the path loss, PEs, FPs, and the number of users can significantly affect the proposed multiuser MRFTHz system.

Linear independence of even periods of modular forms

We show that if the dimension of the space of cuspforms is greater than or equal to three, then any three even periods are linearly independent. We also prove an asymptotic result for an arbitrary number of even periods. These results are achieved by studying the Rankin-Cohen brackets of Eisenstein series.

A unified model specification for sparse and dense functional/longitudinal data

The semi varying-coefficient additive model is a flexible nonparametric regression method, including varying-coefficient model and additive model as its special cases. However, a complex model may lead over-fitting phenomenon, which motivates us to develop a set of testing procedure to judge whether a parsimonious submodel is sufficient or not. Specifically, we propose hypothesis testings to check time-varying property of varying-coefficient component functions and linearity of additive component functions, respectively. For repeated measurements data, it is a subjective choice between sparse and dense case, which may lead to wrong statistical inference. One major contribution of this paper is to introduce consistent testing methodologies in a unified framework of sparse, dense and ultra dense cases of the data, which avoids a subjective choice of data types in practical applications. Extensive Monte Carlo simulation studies investigating the finite sample performance of the proposed methodologies confirm our asymptotic results. We further illustrate our methodologies via two real-life data applications.

Scalar-on-Function Relative Error Regression for Weak Dependent Case

Analyzing the co-variability between the Hilbert regressor and the scalar output variable is crucial in functional statistics. In this contribution, the kernel smoothing of the Relative Error Regression (RE-regression) is used to resolve this problem. Precisely, we use the relative square error to establish an estimator of the Hilbertian regression. As asymptotic results, the Hilbertian observations are assumed to be quasi-associated, and we demonstrate the almost complete consistency of the constructed estimator. The feasibility of this Hilbertian model as a predictor in functional time series data is discussed. Moreover, we give some practical ideas for selecting the smoothing parameter based on the bootstrap procedure. Finally, an empirical investigation is performed to examine the behavior of the RE-regression estimation and its superiority in practice.

Endogeneity in semiparametric threshold regression models with two threshold variables

ABSTRACT This article considers a semiparametric threshold regression model with two threshold variables. The proposed model allows endogenous threshold variables and endogenous slope regressors. Under the diminishing threshold effects framework, we derive consistency and asymptotic results of our proposed estimator for weakly dependent data. We study the finite sample performance of our proposed estimator via small Monte Carlo simulations and apply our model to classify economic growth regimes based on both national public debt and national external debt.

Nonparametric estimation of the shape functions and related asymptotic results

Arriaza et al (Metrika 82:99–124, 2019) introduced the right and left shape functions, which enjoy interesting properties in terms of describing the global form of a distribution. This paper proposes and studies nonparametric estimators of those functions. The estimators involve nonparametric estimation of the quantile and density functions. Pointwise and uniform consistency are proved under general regularity assumptions, as well as the limit in law. Simulations are included to study the practical performance of the proposed estimators. The analysis of a real data set illustrates the methodology.

Asymptotics of discrete q-Freud orthogonal polynomials from the q-Riemann Hilbert problem

We investigate a Riemann–Hilbert problem (RHP), whose solution corresponds to a group of q-orthogonal polynomials studied earlier by Ismail et al. Using RHP theory we determine new asymptotic results in the limit as the degree of the polynomials approach infinity. The RHP formulation also enables us to obtain further properties. In particular, we consider how the class of polynomials and their asymptotic behaviours change under translations of the q-discrete lattice and determine the asymptotics of a related q-Painlevé equation.

Interface behaviour of the slow diffusion equation with strong absorption: Intermediate-asymptotic properties

AbstractThe dynamics of interfaces in slow diffusion equations with strong absorption are studied. Asymptotic methods are used to give descriptions of the behaviour local to a comprehensive range of possible singular events that can occur in any evolution. These events are: when an interface changes its direction of propagation (reversing and anti-reversing), when an interface detaches from an absorbing obstacle (detaching), when two interfaces are formed by film rupture (touchdown) and when the solution undergoes extinction. Our account of extinction and self-similar reversing and anti-reversing is built upon previous work; results on non-self-similar reversing and anti-reversing and on the various types of detachment and touchdown are developed from scratch. In all cases, verification of the asymptotic results against numerical solutions to the full PDE is provided. Self-similar solutions, both of the full equation and of its asymptotic limits, play a central role in the analysis.