Asymptotic Properties Research Articles

DsubCox: a fast subsampling algorithm for Cox model with distributed and massive survival data.

To ensure privacy protection and alleviate computational burden, we propose a fast subsmaling procedure for the Cox model with massive survival datasets from multi-centered, decentralized sources. The proposed estimator is computed based on optimal subsampling probabilities that we derived and enables transmission of subsample-based summary level statistics between different storage sites with only one round of communication. For inference, the asymptotic properties of the proposed estimator were rigorously established. An extensive simulation study demonstrated that the proposed approach is effective. The methodology was applied to analyze a large dataset from the U.S. airlines.

Estimation and Variable Selection for Interval-Censored Failure Time Data with Random Change Point and Application to Breast Cancer Study

Motivated by a breast cancer study, we consider regression analysis of interval-censored failure time data in the presence of a random change point. Although a great deal of literature on interval-censored data has been established, there does not seem to exist an established method that can allow for the existence of random change points. Such data can occur in, for example, clinical trials where the risk of a disease may dramatically change when some biological indexes of the human body exceed certain thresholds. To fill the gap, we will first consider regression analysis of such data under a class of linear transformation models and provide a sieve maximum likelihood estimation procedure. Then a penalized method is proposed for simultaneous estimation and variable selection, and the asymptotic properties of the proposed method are established. An extensive simulation study is conducted and indicates that the proposed methods work well in practical situations. The approaches are applied to the real data from the breast cancer study mentioned above. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Penalized Exponentially Tilted Likelihood for Growing Dimensional Models with Missing Data

This paper develops a penalized exponentially tilted (ET) likelihood to simultaneously estimate unknown parameters and select variables for growing dimensional models with missing response at random. The inverse probability weighted approach is employed to compensate for missing information and to ensure the consistency of parameter estimators. Based on the penalized ET likelihood, we construct an ET likelihood ratio statistic to test the contrast hypothesis of parameters. Under some wild conditions, we obtain the consistency, asymptotic properties, and oracle properties of parameter estimators and show that the constrained penalized ET likelihood ratio statistic for testing the contrast hypothesis possesses the Wilks’ property. Simulation studies are conducted to validate the finite sample performance of the proposed methodologies. Thyroid data taken from the First People’s Hospital of Yunnan Province is employed to illustrate the proposed methodologies.

Asymptotic Properties of Some Freud Polynomials

Asymptotic Properties of Some Freud Polynomials

Asymptotic behavior of bootstrapped extreme order statistics under unknown power normalizing constants

Bootstrapping is a powerful statistical technique, but its reliability for analyzing extreme events with unknown power normalization remains unclear. This paper addresses this issue by exploring the asymptotic properties of bootstrapped extreme order statistics under unknown power normalizing constants. Different estimators of the power normalizing constants are considered. The consistency properties of bootstrapped extreme order statistics with the estimated power normalizing constants are investigated. An extensive simulation study is conducted to identify an optimal bootstrap sample size under power and linear normalization. This study is based on how closely the bootstrapped extreme order statistics align with their theoretical asymptotic limits.

A Note on an Asymptotic Property of the Convexity of Norms

We observe that the recently introduced property local asymptotically midpoint uniform convexity (LAMUC}) is isomorphically distinct from AMUC. Specifically, we show that $$c_0$$ or $$T^*$$, with any locally uniformly rotund norm are strictly locally AMUC. We introduce local AUC in the same spirit and give an example of locally AMUC, not locally AUC and not isomorphic to AMUC space. Following the proof in [10] we observe that these local properties are stable under finite sums with outer space having 1-unconditional basis and uniformly monotone norm.

Group sequential methods to compare receiver operating characteristic curves for correlated and clustered data

It is ethical and cost efficient to sequentially estimate and compare clustered ROC curves. For i.i.d. data, the properties of a single sequential empirical ROC curve has been rigorously studied in Koopmeiners and Feng [Asymptotic properties of the sequential empirical roc, ppv and npv curves under case-control sampling. Ann Stat. 2011;39(6):3234–3261]. When markers are measured on multiple locations of the same study subjects yielding correlated and clustered data, the properties of the sequential empirical ROC curves are unknown. In this setting, we investigate the properties of ROC curves in the group sequential design in the current manuscript. Specifically, we derive the joint asymptotic distribution of two sequential empirical ROC curves at any false positive rate. The theoretical properties allow one to design group sequential designs, taking into account appropriate power analysis. We illustrate our methods using data from the glaucoma diagnostic trial.

Phase-Type Distributions for Sieve Estimation

In many semiparametric models, the infinite-dimensional parameter of direct interest is a probability density, but its nonparametric estimation is usually difficult in the presence of incomplete data. To address this issue, this study promotes phase-type distributions as a method of sieve. Phase-type distributions are dense in the space of nonnegative distributions, closed under minimum, maximum, and convolution, and compatible with the accelerated failure time model. This renders them attractive for sieve density estimation for problems with sophisticated missing data. However, the class of phase-type distributions is over-parameterized, and its approximation error rate for a given density as the number of phases increases remains unknown. These handicaps hinder its theoretical development as a sieve method. In this paper, we design a sieve class of identifiable phase-type densities and establish its approximation error rate for a given density, which is the first error rate result known for phase type distributions. The proposed sieve is then used for semiparametric M-estimation, where the nonparametric component is a density. Building on the approximation error rate results, we establish general asymptotic properties of the phase-type sieve estimators and apply them to models that have complicated data missingness and cannot be easily handled by existing methods. Our simulation and real data analysis focus on right-censored data with missing indicators, in which we demonstrate that our estimators are more efficient than existing estimators.

Assumption-lean regression for non-negative response via variational representation

We present a novel estimator for regression parameters that does not require any assumptions about the conditional distribution family of the response variable, except for non-negativity. This estimator is particularly useful for handling semi-continuous, zero-inflated, and proportional data, eliminating the need for complex models and simplifying the inference process. The estimator is unaffected by over-under dispersion, heavy tails or other problems related to the incorrect assumptions about the response distribution. In addition, the asymptotic properties of the estimator do not depend on the underlying conditional distribution. Apart from that, our method does not involve kernel-based estimation and is, therefore, free from issues related to smoothing parameter selection. The R code for implementing the method and reproducing the results is available in the supplementary materials

Highly accurate real-space electron densities with neural networks.

Variational abinitio methods in quantum chemistry stand out among other methods in providing direct access to the wave function. This allows, in principle, straightforward extraction of any other observable of interest, besides the energy, but, in practice, this extraction is often technically difficult and computationally impractical. Here, we consider the electron density as a central observable in quantum chemistry and introduce a novel method to obtain accurate densities from real-space many-electron wave functions by representing the density with a neural network that captures known asymptotic properties and is trained from the wave function by score matching and noise-contrastive estimation. We use variational quantum Monte Carlo with deep-learning Ansätze to obtain highly accurate wave functions free of basis set errors and from them, using our novel method, correspondingly accurate electron densities, which we demonstrate by calculating dipole moments, nuclear forces, contact densities, and other density-based properties.

K-Nearest Neighbour Estimation of the Conditional Set-Indexed Empirical Process for Functional Data: Asymptotic Properties

k-Nearest Neighbour Estimation of the Conditional Set-Indexed Empirical Process for Functional Data: Asymptotic Properties

Change point detection for clustered survival data with marginal proportional hazards model

In survival analysis, models with a change point for independent survival data have been extensively studied. However, few works in the literature focus on clustered survival data with a change point, despite the prevalence of such survival data in clinical research. In this article, we propose the maximal score test, along with its normalized version, to detect the existence of change point effects for the marginal proportional hazards model. We establish the asymptotic distributions of the proposed test statistics under both the null and local alternative hypotheses. To approximate the critical points for the proposed tests, a simple Monte Carlo method is presented. Furthermore, when the null hypothesis of no change point effects is rejected, we propose maximum smoothed partial likelihood estimators for the unknown change point and the regression parameters and derive the asymptotic properties of the estimators. Extensive simulation studies are carried out to evaluate the finite sample performance of the proposed methods. Finally, we illustrate the proposed methods with applications to the breast cancer data from the Prostate, Lung, Colorectal, and Ovarian Cancer Screening Trial.

Distribution-on-Scalar Single-Index Quantile Regression Model for Handling Tumor Heterogeneity

This article develops a distribution-on-scalar single-index quantile regression modeling framework to investigate the relationship between cancer imaging responses and scalar covariates of interest while tackling tumor heterogeneity. Conventional association analysis methods typically assume that the imaging responses are well-aligned after some preprocessing steps. However, this assumption is often violated in practice due to imaging heterogeneity. Although some distribution-based approaches are developed to deal with this heterogeneity, major challenges have been posted due to the nonlinear subspace formed by the distributional responses, the unknown nonlinear association structure, and the lack of statistical inference. Our method can successfully address all the challenges. We establish both estimation and inference procedures for the unknown functions in our model. The asymptotic properties of both estimation and inference procedures are systematically investigated. The finite-sample performance of our proposed method is assessed by using both Monte Carlo simulations and a real data example on brain cancer images from TCIA-GBM collection.

Analysis of fractional Black-Scholes, Ornstein-Uhlenbeck, and Langevin models: a minimum distance estimation approach

The present research investigates parameter estimation in fractional stochastic models, specifically the Fractional Black-Scholes, Fractional Ornstein-Uhlenbeck, and Fractional Langevin models. These models, driven by fractional Brownian motion (fBm), capture long-memory effects and complex dependencies often observed in financial and physical systems. To efficiently estimate the parameters of these models, we apply the Minimum Distance Estimation (MDE) method. This approach tackles the challenges posed by long-range dependence, ensuring both asymptotic consistency and normality. We analyze the asymptotic properties of the estimators, including their consistency, asymptotic normality, and efficiency. By providing a unified theoretical framework, the current research highlights the robustness and applicability of the MDE method for parameter inference in fractional models. The findings contribute significantly to diverse fields such as quantitative finance, econometrics, and statistical physics.

Asymptotic behavior of the equivalent Timoshenko linear beam model used in the analysis of tower buildings

We study the asymptotic properties of a continuous Timoshenko linear beam model immersed in a three-dimensional space and used in the analysis of tower buildings. Assume that the bending and axial behaviors are coupled on the one hand, while the shear and torsional behaviors are coupled on the other hand. If the displacement vector is totally damped and the rotational vector is undamped, or if the rotational vector is totally damped and the displacement vector is undamped, then the system is proved to be exponentially stable, under some assumptions on the physical parameters involved in the problem. We also consider the case when the bending and axial behaviors and/or the shear and torsional behaviors are uncoupled. In this situation, the stability properties are different if the rotational or the displacement vector is damped or not: the system can be exponentially stable, polynomially stable or even unstable.

Asymptotic and Oscillatory Properties for Even-Order Nonlinear Neutral Differential Equations with Damping Term

This research focuses on studying the asymptotic and oscillatory behavior of a special class of even-order nonlinear neutral differential equations, including damping terms. The research aims to achieve qualitative progress in understanding the relationship between the solutions of these equations and their associated functions. Leveraging the symmetry between positive and negative solutions simplifies the derivation of criteria that ensure the oscillation of all solutions. Using precise techniques such as the Riccati method and comparison methods, innovative criteria are developed that guarantee the oscillation of all the solutions of the studied equations. The study provides new conditions and effective analytical tools that contribute to deepening the theoretical understanding and expanding the practical applications of these systems. Based on solid scientific foundations and previous studies, the research concludes with the presentation of examples that illustrate the practical impact of the results, highlighting the theoretical value of research in the field of neutral differential equations.

Automatic Threshold Selection for Generalized Pareto and Pareto–Poisson Distributions in Rainfall Analysis: A Case Study Using the NOAA NCDC Daily Rainfall Database

Both extreme-excess modeling and extreme-value analysis of precipitation events frequently utilize the Generalized Pareto (GP) distribution to model peaks above a selected threshold. However, selecting an appropriate threshold remains a complex and challenging task, which has discouraged many practitioners from employing Pareto or Pareto–Poisson distributions for extreme-value analysis. Recent analyses of threshold selection methods proposed in the technical literature, particularly when applied to rainfall records with high quantization levels, have shown that nonparametric methods are often unreliable. Additionally, methods relying on the asymptotic properties of the GP distribution tend to produce unrealistically high threshold estimates. In contrast, graphical methods and goodness-of-fit (GoF) metrics that account for the pre-asymptotic behavior of the GP distribution have demonstrated better performance. Despite these improvements, there remains no automatic and statistically robust methodology for threshold selection. This study develops an automatic, statistically sound procedure for optimal threshold selection, leveraging weighted mean square errors and internally studentized residuals. The proposed method outperforms existing approaches in terms of accuracy, as demonstrated through numerical experiments and its application to real-world data from the NOAA NCDC Daily Rainfall Database. Results indicate that the method not only improves threshold estimation precision but also enhances the reliability of extreme-value analysis for precipitation records, making it a valuable tool for hydrological applications. The findings emphasize the practical implications of the method for analyzing extreme rainfall events and its potential for broader climatological studies.

Smoothing Estimation of Parameters in Censored Quantile Linear Regression Model

In this paper, we propose a smoothing estimation method for censored quantile regression models. The method associates the convolutional smoothing estimation with the loss function, which is quadratically derivable and globally convex by using a non-negative kernel function. Thus, the parameters of the regression model can be computed by using the gradient-based iterative algorithm. We demonstrate the convergence speed and asymptotic properties of the smoothing estimation for large samples in high dimensions. Numerical simulations show that the smoothing estimation method for censored quantile regression models improves the estimation accuracy, computational speed, and robustness over the classical parameter estimation method. The simulation results also show that the parametric methods perform better than the KM method in estimating the distribution function of the censored variables. Even if there is an error setting in the distribution estimation, the smoothing estimation does not fluctuate too much.

Positive Periodic Solutions of Non-Autonomous Predator-Prey System with Stage-Structured Predator on Time Scales

In this note, we investigate the existence and asymptotic property of positive periodic solutions to non-autonomous predator-prey system with stage-structured predator on time scales. Via Schauder’s fixed theorem, easily verifiable sufficient existence conditions of positive periodic solutions for the considered system are obtained. We also study asymptotic property of positive periodic solutions on the basis of existence conditions. Due to the symmetry of periodic solutions, the results of this paper have a certain impact on the study of symmetry. It should be pointed out that the system we are studying is built on arbitrary time scale, so our results generalize the results of existing continuous or discrete systems. Furthermore, we develop Schauder’s fixed theorem for studying the delay system on time scales.

A nonparametric Bootstrap CUSUM multi-chart for detecting unknown abrupt changes

This article considers the online monitoring of changes using a nonparametric Bootstrap approach. Since pre-change historical observations can influence the performance of change detection, we employ the Bootstrap resampling technique in the nonparametric CUSUM multi-chart to reduce the estimation error brought by pre-change historical observations. We also present the asymptotic properties of the Bootstrap approach. Additionally, we use simulated data to present the relevant results. Moreover, we apply the method in this paper to network data, which demonstrates the potential advantages of the proposed method in application.