Asymptotic Properties Research Articles

Multi-pole soliton of discrete integrable equations and modified Riemann-Hilbert approach: discrete Hirota equation

In this study, the Riemann-Hilbert approach was developed and applied to the discrete Hirota equation. We constructed a modified Riemann-Hilbert problem compatible with the discrete Hirota equation and derived a reconstruction formula for its solutions. Because the characteristic function contains a potential, we modify the Riemann-Hilbert approach to make the Riemann-Hilbert matrix have good asymptotic properties. We believe that the modified Riemann-Hilbert approach can also be applied to other discrete integrable models. By using the direct method of Laurent series, we obtained the expression of multi-pole solutions for the discrete Hirota equation and demonstrated the dynamic behavior of some solutions.

Unified specification tests in partially linear quantile regression models

We propose specification tests for parametric quantile regression models versus semiparametric alternatives over a continuum of quantile levels. The test statistics are constructed as continuous functionals of a quantile-marked residual process. We show that using an orthogonal projection on the tangent space of nuisance parameters at each quantile index delivers unified asymptotic properties for tests based on different estimators. Consistency of the tests and asymptotic power under a sequence of local alternatives converging to the null at a parametric rate are also discussed. We propose a simple multiplier bootstrap procedure to carry out the tests, whose nominal levels are well approximated in our simulation study for modest sample sizes.

A New Family of Distributions Based on Amalgamation of Two Methods with an Application to the Rayleigh Model

In this paper, we present the Alpha power generalized odd generalized exponential-G (APGOGE-G) family of distributions and provides the most common shapes of the hazard rate function: increasing, decreasing, bathtub, and inverted bathtub. We provide some of its structural properties. We estimate the parameters by maximum likelihood estimation method, and perform a simulation study to verify the asymptotic properties of the estimator for the inverse Weibull baseline. The practicality of the new APGOGE-Rayleigh model is shown through application to uncensored real dataset.

Asymptotic analysis of two-dimensional Oldroyd fluids in unbounded domains: Global attractors and their dimension

In this work, we consider the two-dimensional Oldroyd model for the non-Newtonian fluid flows (viscoelastic fluid) in Poincaré domains (bounded or unbounded) and study their asymptotic behavior. We establish the existence of a global attractor in Poincaré domains using asymptotic compactness property. Since the high regularity of solutions is not easy to establish, we prove the asymptotic compactness of the solution operator by applying Kuratowski’s measure of noncompactness, which relies on uniform-tail estimates and the flattening property of the solution. Finally, the estimates for the Hausdorff as well as fractal dimensions of global attractors are also obtained.

Receiver operating characteristic analysis for paired comparison data

Abstract Paired comparison models are used for analysing data that involves pairwise comparisons among a set of objects. When the outcomes of the pairwise comparisons have no ties, the paired comparison models can be generalized as a class of binary response models. Receiver operating characteristic (ROC) curves and their corresponding areas under the curves are commonly used as performance metrics to evaluate the discriminating ability of binary response models. Despite their individual wide range of usage and their close connection to binary response models, ROC analysis to our knowledge has never been extended to paired comparison models since the problem of using different objects as the reference in paired comparison models prevents traditional ROC approach from generating unambiguous and interpretable curves. We focus on addressing this problem by proposing two novel methods to construct ROC curves for paired comparison data which provide interpretable statistics and maintain desired asymptotic properties. The methods are then applied and analysed on head-to-head professional sports competition data.

Robust Recovery of Optimally Smoothed Polymer Relaxation Spectrum from Stress Relaxation Test Measurements.

The relaxation spectrum is a fundamental viscoelastic characteristic from which other material functions used to describe the rheological properties of polymers can be determined. The spectrum is recovered from relaxation stress or oscillatory shear data. Since the problem of the relaxation spectrum identification is ill-posed, in the known methods, different mechanisms are built-in to obtain a smooth enough and noise-robust relaxation spectrum model. The regularization of the original problem and/or limit of the set of admissible solutions are the most commonly used remedies. Here, the problem of determining an optimally smoothed continuous relaxation time spectrum is directly stated and solved for the first time, assuming that discrete-time noise-corrupted measurements of a relaxation modulus obtained in the stress relaxation experiment are available for identification. The relaxation time spectrum model that reproduces the relaxation modulus measurements and is the best smoothed in the class of continuous square-integrable functions is sought. Based on the Hilbert projection theorem, the best-smoothed relaxation spectrum model is found to be described by a finite sum of specific exponential-hyperbolic basis functions. For noise-corrupted measurements, a quadratic with respect to the Lagrange multipliers term is introduced into the Lagrangian functional of the model's best smoothing problem. As a result, a small model error of the relaxation modulus model is obtained, which increases the model's robustness. The necessary and sufficient optimality conditions are derived whose unique solution yields a direct analytical formula of the best-smoothed relaxation spectrum model. The related model of the relaxation modulus is given. A computational identification algorithm using the singular value decomposition is presented, which can be easily implemented in any computing environment. The approximation error, model smoothness, noise robustness, and identifiability of the polymer real spectrum are studied analytically. It is demonstrated by numerical studies that the algorithm proposed can be successfully applied for the identification of one- and two-mode Gaussian-like relaxation spectra. The applicability of this approach to determining the Baumgaertel, Schausberger, and Winter spectrum is also examined, and it is shown that it is well approximated for higher frequencies and, in particular, in the neighborhood of the local maximum. However, the comparison of the asymptotic properties of the best-smoothed spectrum model and the BSW model a priori excludes a good approximation of the spectrum in the close neighborhood of zero-relaxation time.

Nonstationary longitudinal autoregressive mixed model for count data with measurement error in covariates: Estimation and asymptotics

Several methods have been proposed in the literature for computing unbiased and efficient estimates of the parameters of generalized linear models when the covariates are measured with error. However, to our knowledge, no documented research on computational techniques for parameter estimation currently exist in the literature when the data is a longitudinal count data influenced by an unobservable latent variable and observable covariates that are measured with error. In this paper, we propose a nonstationary conditionally Poisson mixed model for such data and develop unbiased estimating equations with iterative methods for computing estimates of the effect of the covariates, variance of the latent variable, and the correlation index parameter. The performance of the iterative methods is examined through extensive simulation studies. The results show that the methods performed well when the magnitude of the measurement error is not so large as to dominate or mask the effect of the true covariates. Using observed longitudinal count data on the number of patents awarded to 168 firms in the United States from 1974 to 1979 along with associated covariate information on the type of firm, log of the book value of capital in 1972 and research and development (R & D) expenditures we have demonstrated how the methods proposed in this paper can be applied to a real data. In addition, we derive the influence function of the estimator of the covariate effect and discuss the asymptotic properties of the estimator. Journal of Statistical Research 2024, Vol. 58, No. 1, pp. 151-180.

Linear-quadratic Tobit regression model with a change point due to a covariate threshold

This paper considers a linear-quadratic Tobit regression model, which is developed for modelling the mixture structure with a line segment and a quadratic segment intersecting at an unknown change point. Due to the smoothness of such model structure, the regression coefficients and the change point can be obtained by directly maximising the likelihood function. The proposed estimator has rapid convergence rate and high estimation accuracy, without other computation burden. The asymptotic properties for the proposed estimator are derived by using empirical process theory. A sup-likelihood ratio test procedure is developed for testing the existence of a change point, and its limiting distributions are derived. The good finite sample performances of the proposed estimator are illustrated by numerical studies and empirical applications to the MGUS and GDP data.

Limiting distributions for particles near the frontier of spatially inhomogeneous branching symmetric α-stable processes

We investigate the evolution of particles near the frontier and determine the limiting distribution of these particles, for the branching symmetric α-stable processes. Here, we assume that the branching rate measure is the compactly supported Kato class measure. This process is characterized by the Schrödinger-type operator. Then, these asymptotic properties are determined by the characteristic quantities such as the eigenvalue and eigenfunction of it. By using these, we prove limit theorems. This research is an extension of Nishimori [Limiting distributions for particles near the frontier of spatially inhomogeneous branching Brownian motions. Acta Appl. Math. 184(10) (2023)] on the branching Brownian motions.

Inference for restricted mean survival time as a function of restriction time under length-biased sampling.

The restricted mean survival time (RMST) is often of direct interest in clinical studies involving censored survival outcomes. It describes the area under the survival curve from time zero to a specified time point. When data are subject to length-biased sampling, as is frequently encountered in observational cohort studies, existing methods cannot estimate the RMST for various restriction times through a single model. In this article, we model the RMST as a continuous function of the restriction time under the setting of length-biased sampling. Two approaches based on estimating equations are proposed to estimate the time-varying effects of covariates. Finally, we establish the asymptotic properties for the proposed estimators. Simulation studies are performed to demonstrate the finite sample performance. Two real-data examples are analyzed by our procedures.

Functional index coefficient models for locally stationary time series

In the analysis of nonlinear time series, we propose a novel functional index coefficient model for the locally stationary data. The proposed model can effectively capture the dynamic interaction effects between variables and the nature of data evolution. Drawing the idea from the spline-backfitted method, we propose a three-step estimation procedure and establish the asymptotic properties of the resulting nonparametric estimators. We further construct simultaneous confidence bands for the time-varying functions to explore the global variation of the original data. We also develop a test statistic to check the time-varying properties based on a bootstrap procedure. Simulation studies have been conducted to investigate the finite sample performance of the proposed methods. Two real applications in the finance market and the Hong Kong respiratory and circulatory disease data are analysed for illustration.

A New Mixture Model With Cure Rate Applied to Breast Cancer Data.

We introduce a new modelling for long-term survival models, assuming that the number of competing causes follows a mixture of Poisson and the Birnbaum-Saunders distribution. In this context, we present some statistical properties of our model and demonstrate that the promotion time model emerges as a limiting case. We delve into detailed discussions of specific models within this class. Notably, we examine the expected number of competing causes, which depends on covariates. This allows for direct modeling of the cure rate as a function of covariates. We present an Expectation-Maximization (EM) algorithm for parameter estimation, to discuss the estimation via maximum likelihood (ML) and provide insights into parameter inference for this model. Additionally, we outline sufficient conditions for ensuring the consistency and asymptotic normal distribution of ML estimators. To evaluate the performance of our estimation method, we conduct a Monte Carlo simulation to provide asymptotic properties and a power study of LR test by contrasting our methodology against the promotion time model. To demonstrate the practical applicability of our model, we apply it to a real medical dataset from a population-based study of incidence of breast cancer in São Paulo, Brazil. Our results illustrate that the proposed model can outperform traditional approaches in terms of model fitting, highlighting its potential utility in real-world scenarios.

On the asymptotic normality of trimmed and winsorized L-statistics

. There are several ways to establish the asymptotic normality of L-statistics, which depend on the choice of the weights-generating function and the cumulative distribution selection of the underlying model. In this study, we focus on establishing computational formulas for the asymptotic variance of two robust L-estimators: the method of trimmed moments (MTM) and the method of winsorized moments (MWM). We demonstrate that two asymptotic approaches for MTM are equivalent for a specific choice of the weights-generating function. These findings enhance the applicability of these estimators across various underlying distributions, making them effective tools in diverse statistical scenarios. Such scenarios include actuarial contexts, such as payment-per-payment and payment-per-loss data scenarios, as well as in evaluating the asymptotic distributional properties of distortion risk measures. The effectiveness of our methodologies depends on the availability of the cumulative distribution function, ensuring broad usability in various statistical environments.

A novel test of missing completely at random: U-statistics-based approach

In this paper, a novel test for testing whether data are missing completely at random is proposed. Asymptotic properties of the test are derived utilizing the theory of non-degenerate U-statistics. It is shown that the novel test statistic coincides with the well-known Little's d 2 statistic in the case of a multivariate data that has only one variable susceptible to missingness. Then, the extensive simulation study is conducted to examine the performance of the test in terms of the preservation of type I error and in terms of power. Various underlying distributions, dimensions of the data, sample sizes and alternatives are used. Performance of the Little's MCAR test is used as a benchmark for the comparison. The novel test shows better performance in all of the studied scenarios, better preserving the type I error and having higher empirical powers for every studied alternative.

Exogeneity tests and weak identification in IV regressions: Asymptotic theory and point estimation

This paper provides new insights on exogeneity tests in linear IV models and their use for estimation, when identification fails or may not be strong. We make two main contributions. First, we show that Durbin-Wu-Hausman (DWH) and Revankar-Hartley (RH) exogeneity tests have correct level asymptotically, even when the first-stage coefficient matrix (which controls identification) is rank-deficient. We provide necessary and sufficient conditions under which these tests are consistent. In particular, we show that test consistency can hold even when identification fails, provided at least one component of the structural parameter vector is identifiable. Second, we study point estimation after estimator (or model) selection, when the outcome of a DWH/RH test determines whether OLS or an IV method is employed in the second-stage. For this purpose, we use (non-local) concepts of asymptotic bias, asymptotic mean squared error (AMSE), and asymptotic relative efficiency (ARE), which remain applicable even when the estimators considered do not have moments (as can happen for 2SLS) or may be inconsistent. We study the asymptotic properties of OLS, 2SLS, and pretest estimators which select OLS or 2SLS based on the outcome of a DWH/RH test. We show that: (i) OLS typically dominates 2SLS estimator asymptotically for MSE across a broad spectrum of cases, including weak identification and moderate endogeneity; (ii) exogeneity-pretest estimators exhibit consistently good performance and asymptotically dominate both OLS and 2SLS. The proposed theoretical findings are documented by Monte Carlo simulations.

Asymptotic smoothness effects and global attractor for a peridynamic model with energy damping

AbstractIn this paper, we consider a peridynamic model with energy damping inspired by the works of Balakrishnan and Taylor on “damping models” based on the instantaneous total energy of the system. We study the asymptotic behavior of solutions, in the sense of attractors, of these peridynamic models in suitable phase space; more precisely, we prove a result of existence and characterization of compact global attractors with a nonlinear strongly continuous semigroup approach based in the asymptotic smoothness property thanks to Chueshov and Lasiecka and Nakao's lemma.

Tail variance and confidence of using tail conditional expectation: analytical representation, capital adequacy, and asymptotics

Abstract In this paper, we explore the applications of Tail Variance (TV) as a measure of tail riskiness and the confidence level of using Tail Conditional Expectation (TCE)-based risk capital. While TCE measures the expected loss of a risk that exceeds a certain threshold, TV measures the variability of risk along its tails. We first derive analytical formulas of TV and TCE for a large variety of probability distributions. These formulas are useful instruments for relevant research works on tail risk measures. We then propose a distribution-free approach utilizing TV to estimate the lower bounds of the confidence level of using TCE-based risk capital. In doing so, we introduce sharpened conditional probability inequalities, which halve the bounds of conventional Markov and Cantelli inequalities. Such an approach is easy to implement. We further investigate the characterization of tail risks by TV through an exploration of TV’s asymptotics. A distribution-free limit formula is derived for the asymptotics of TV. To further investigate the asymptotic properties, we consider two broad distribution families defined on tails, namely, the polynomial-tailed distributions and the exponential-tailed distributions. The two distribution families are found to exhibit an asymptotic equivalence between TV and the reciprocal square of the hazard rate. We also establish asymptotic relationships between TCE and VaR for the two families. Our asymptotic analysis contributes to the existing research by unifying the asymptotic expressions and the convergence rate of TV for Student-t distributions, exponential distributions, and normal distributions, which complements the discussion on the convergence rate of univariate cases in [28]. To show the usefulness of our results, we present two case studies based on real data from the industry. We first show how to use conditional inequalities to assess the confidence of using TCE-based risk capital for different types of insurance businesses. Then, for financial data, we provide alternative evidence for the relationship between the data frequency and the tail categorization by the asymptotics of TV.

Adjusted Design Effect Model for Longitudinal Survey Data

In this research, we propose a longitudinal adjusted design effect model that integrates both the design effect and time effect, offering reliable estimators of variance. We investigate the asymptotic properties of certain types of estimators. Through simulation studies, we show the exceptional performance of the proposed method, demonstrating its superiority in producing accurate standard error estimators compared to existing approaches.

On Asymptotic Properties of Solutions for Differential Equations of Neutral Type

On Asymptotic Properties of Solutions for Differential Equations of Neutral Type

General estimation results for tdVARMA array models

The article will focus on vector autoregressive‐moving average (VARMA) models with time‐dependent coefficients (td) to represent general nonstationary time series, not necessarily Gaussian. The coefficients depend on time, possibly on the length of the series , hence the name tdVARMA for the models, but not necessarily on the rescaled time . As a consequence of the dependency on of the model, we need to consider array processes instead of stochastic processes. Under appropriate assumptions, it is shown that a Gaussian quasi‐maximum likelihood estimator is consistent in probability and asymptotically normal. The theoretical results are illustrated using three examples of bivariate processes, the first two with marginal heteroscedasticity. The first example is a tdVAR(1) process while the second example is a tdVMA(1) process. In these two cases, the finite‐sample behavior is checked via a Monte Carlo simulation study. The results are compatible with the asymptotic properties even for small . A third example shows the application of the tdVARMA models for a real time series.