Asymptotic Distribution Research Articles

Likelihood Ratio Tests in Random Graph Models with Increasing Dimensions

We explore the Wilks phenomena in two random graph models: the β -model and the Bradley–Terry model. For two increasing dimensional null hypotheses, including a specified null H 0 : β i = β i 0 for i = 1 , … , r and a homogenous null H 0 : β 1 = ⋯ = β r , we reveal high dimensional Wilks’ phenomena that the normalized log-likelihood ratio statistic, [ 2 { l ( β ̂ ) − l ( β ̂ 0 ) } − r ] / ( 2 r ) 1 / 2 , converges in distribution to the standard normal distribution as r goes to infinity. Here, l ( β ) is the log-likelihood function on the model parameter β = ( β 1 , … , β n ) ⊤ , β ̂ is its maximum likelihood estimator (MLE) under the full parameter space, and β ̂ 0 is the restricted MLE under the null parameter space. For the homogenous null with a fixed r, we establish Wilks-type theorems that 2 { l ( β ̂ ) − l ( β ̂ 0 ) } converges in distribution to a chi-square distribution with r − 1 degrees of freedom, as the total number of parameters, n, goes to infinity. When testing the fixed dimensional specified null, we find that its asymptotic null distribution is a chi-square distribution in the β -model. However, unexpectedly, this is not true in the Bradley–Terry model. By developing several novel technical methods for asymptotic expansion, we explore Wilks-type results in a principled manner; these principled methods should be applicable to a class of random graph models beyond the β -model and the Bradley–Terry model. Simulation studies and real network data applications further demonstrate the theoretical results. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Influence function-based empirical likelihood for area under the receiver operating characteristic curve in presence of covariates.

In receiver operating characteristicROC analysis, the area under the ROC curve (AUC) is a popular one number summary of the discriminatory accuracy of a diagnostic test. AUC measures the overall diagnostic accuracy of a test but fails to account for the effect of covariates when covariates are present and associated with the test results. Adjustment for covariate effects can greatly improve the diagnostic accuracy of a test. In this paper, using information provided by the influence function, empirical likelihood (EL) methods are proposed for inferences of AUC in presence of covariates. For parameters in the AUC regression model, it is shown that the asymptotic distribution of the influence function-based empirical log-likelihood ratio statistic is a standard chi-square distribution. Hence, confidence regions for the regression parameters can be obtained without any variance estimation. Simulation studies are conducted to compare the finite sample performances of the proposed EL based methods with the existing normal approximation (NA) based method in the AUC regression. Simulation results indicate that the bootstrap-calibrated influence function-based empirical likelihood (BIFEL ) confidence region outperforms the NA-based confidence region in terms of coverage probability. We also propose an interval estimation method for the covariate-adjusted AUC based on the BIFEL confidence region. Finally, we illustrate the recommended method with a real prostate-specific antigen data example.

A Quality Decision-Making Approach based on the Asymptotic Distribution of the Taguchi Index Estimator

A Quality Decision-Making Approach based on the Asymptotic Distribution of the Taguchi Index Estimator

Goodness of fit tests in spatial autoregressive stochastic frontier models

. This article mainly considers goodness of fit tests for the inefficiency in spatial autoregressive stochastic frontier (SARSF) models by utilizing the characteristic function of the centralized composite error term. As a prerequisite of the tests, we propose easy-to-compute moment estimators for distributional parameters of such a term, showing the consistency based on spatial near-epoch dependent properties. Meanwhile, the asymptotic distribution of the estimators is derived by generalizing the central limit theorem (CLT) in Kelejian and Prucha (2001) from linear-quadratic forms to polynomial-trigonometric forms. Then, the cosine and sine tests, being simple to implement in SARSF analysis, are established to explore the distributional structure of the centered error. With the help of the generalized CLT, both trigonometry statistics are proved to follow an asymptotically chi-square distribution. Moreover, Monte Carlo simulation is conducted to investigate the finite sample performance of the moment estimators and trigonometry tests. Finally, in the efficiency analysis of Chinese A-shared listed companies in 2005, our tests reject the half-normal and exponential distributions and fail to reject the gamma one. This conforms to our subsequent calculation that the efficiency estimates based on the gamma model potentially lie within a more realistic range, but the other two models lead to significant distortions in efficiency estimates.

Survival Analysis with Graph-Based Regularization for Predictors

Abstract We study the variable selection problem in survival analysis to identify the most important factors affecting survival time. Our method incorporates prior knowledge of mutual correlations among variables, represented through a graph. We utilize the Cox proportional hazard model with a graph-based regularizer for variable selection. We present a computationally efficient algorithm which is developed to solve the graph regularized maximum likelihood problem by establishing connections with the group lasso and provide theoretical guarantees about the recovery error and asymptotic distribution of the proposed estimators. The improved performance of the proposed approach compared with existing methods is demonstrated in both synthetic and real organ transplantation datasets.

A statistical study of energetic particle events associated with interplanetary shocks observed by Solar Orbiter in solar cycle 25

We studied energetic particle intensity profiles observed by Solar Orbiter during the time period from April 2020 to April 2023, associated with the passage of interplanetary (IP) shocks. For our study we considered 58 IP forward shocks and analysed the possible correlations between some IP shock parameters and the electron and proton responses to the passage of the IP shocks. We investigated which shock signatures are more likely related to the efficiency of the IP shocks with respect to particle acceleration. We introduced a variable that characterises the contamination induced by protons in the electron channels of the Electron Proton Telescope (EPT) part of the Energetic Particle Detector (EPD) suite of instruments on board Solar Orbiter, which allowed us to identify the cases in which the intensity time profiles of electrons at energies le240 keV showed a real response at the passage of IP shocks. In the case of protons, we searched for the response in seven energy ranges from 52 keV to 15 MeV, and based on the shape of the proton response at low energies (∼100 keV), we divided the profiles into weak responses, peaks (regular or irregular), plateaus, and unclear responses. For the regular peak and plateau types we constructed an average time profile by applying superposed epoch analysis. For the response in electrons and protons, and for the different types of proton responses at different energies, we analysed the corresponding IP shock parameters, aiming to understand which ones are important to form a certain type of time profile or to achieve a certain energy. We also included a comparison between the proton intensity time profiles in the upstream region and, assuming the predictions of the diffusive shock acceleration (DSA) theory, identified the values of the mean free path in several cases. We found that the IP shock efficiency in the energisation of both electrons and protons is strongly energy dependent. Cases of electron acceleration are rare. Only in about ∼8$%$ of the events for energies le100 keV and in ∼2$%$ for energies le250 keV did the electron intensities show an unambiguous response at the passage of IP shocks (with those accompanied by a response being mainly oblique or quasi-perpendicular). The shocks for which we identified a response in ∼100 keV proton intensity time profiles come to ∼ 83% of the IP shocks under study, and are parallel or quasi-parallel. The ability to accelerate protons to higher energies and to form a particular shape of the particle response to the IP shock passage mostly depends on the IP shock speed. Based on the analysis of time profiles and the occurrence of unambiguous electron acceleration at shocks, the acceleration mechanism behind the electron energisation is unlikely to be DSA, but shock drift acceleration (SDA) remains a candidate for the acceleration mechanism. Proton time profiles of the plateau type around the IP shock front can be achieved with an IP shock speed above 800 km s^-1 and an ambient mean free path ≤ 0.015 au, reproducing the asymptotic steady-state ion distribution reached in the classical DSA solution.

Limit theory for martingale transforms with heavy-tailed noise

A limit theorem for partial sums of martingale transforms with multiplicative noise and ergodic transform processes is established, which leads to regularly varying rates and stable limits. Its establishment is facilitated by the derivation of an extension to empirical distributions of Breiman’s Theorem along with the Principle of Conditioning. The theorem is applied for the development of the limit theory of the least squares estimator in regressions with heavy tailed noise, as well as of the limit theory of the Gaussian QMLE in GARCH-type models. Depending on the index of stability, regularly varying rates and asymptotic stable distributions, or inconsistency, are obtained.

Time‐Varying Dispersion Integer‐Valued GARCH Models

ABSTRACTWe introduce a general class of INteger‐valued Generalized AutoRegressive Conditionally Heteroscedastic (INGARCH) processes by allowing simultaneously time‐varying mean and dispersion parameters. We call such models time‐varying dispersion INGARCH (tv‐DINGARCH) models. More specifically, we consider mixed Poisson INGARCH models and allow for dynamic modeling of both mean and dispersion parameters. We derive conditions to obtain stochastic stability of tv‐DINGARCH processes. Additionally, we study maximum likelihood estimation in detail including its asymptotic distribution. A restricted bootstrap procedure is proposed for testing constant dispersion against time‐varying dispersion. Monte Carlo simulation studies are presented for checking point estimation, standard errors, and the performance of the restricted bootstrap approach. We apply the tv‐DINGARCH process to model the weekly number of reported measles infections in North Rhine‐Westphalia, Germany, from January 2001 to May 2013, and compare its performance to the ordinary INGARCH approach.

Parameter Estimation for Doubly Geometric Process with Inverse Gaussian Distribution and its Applications

The geometric process (GP) has an important role in the reliability theory. Although it is used as a stochastic model in various areas of application, it has some limitations to fit the data purposes of various empirical studies. The doubly geometric process (DGP) has been proposed to overcome these limitations. The parameter estimation problem is very important for both GP and DGP. In this study, the parameter estimation problem for DGP when the distribution of the first interarrival time is assumed to be inverse Gaussian (IG) distribution with parameters [Formula: see text], [Formula: see text] is considered. First, the model parameters are estimated by using the maximum likelihood (ML) method, and then for obtained estimators, asymptotic joint distribution and asymptotic unbiasedness and consistency properties are investigated. A test statistic is introduced based on ML methods to distinguish DGP from GP. The non-parametric (NP) methods which are least square (LSE) and log-LSE are presented. Additionally, modified moment (MM) estimators are obtained. The efficiencies of the ML estimators are compared with MM estimators by an extensive simulation study. It is concluded that the ML estimators perform better than the MM estimators. Finally, three real-life data examples are presented to illustrate the applicability of the method. It is shown that the DGP can model the related data sets.

Validating attribute hierarchies in cognitive diagnosis models.

Cognitive diagnosis models (CDMs) are restricted latent class models that are widely used in educational and psychological fields. Attribute hierarchy, as an important structural feature of the CDM, can provide critical information for inferring examinees' attribute mastery patterns. Previous studies usually formulate likelihood ratio (LR) tests for full models and hierarchical models to validate attribute hierarchies, but their asymptotic distributions tend to become non-standard, resulting in test failures. This study proposes the Wald statistic to statistically test the a priori defined attribute hierarchy. Specifically, two covariance matrix estimators, empirical cross-product information matrix (XPD), and observed information matrix (Obs), are considered to compute the Wald statistic, referred to as Wald-XPD and Wald-Obs, respectively. Simulation studies with various factors were conducted to investigate the performance of the new methods. The results show that Wald-XPD has an acceptable empirical performance with high or low quality items and a higher test efficiency. Real datasets were also analyzed for illustrative purpose.

Aggregated projection method: a new approach for group factor model

Identifying the global factors among grouped data is crucial in the group factor model. In this paper, we propose a novel objective function for the task by maximizing the average of correlations between the latent global factors and group factors, solved through the eigen-decomposition of the aggregated projection matrix. Our method is not only computationally efficient but also robust to strongly correlated local factors. We establish the consistency of the global/local factor number estimation and the consistency and asymptotic distributions of the estimated global/local factors and loadings. Simulation studies show that our method outperforms state-of-the-art methods in various scenarios. The proposed method is applied to analyze the growth rate of house prices in the United States, identifying one global factor that significantly influences national house prices and several local factors that demonstrate the effects of each state.

Parameter estimation for partially observed stochastic processes driven by Ornstein–Uhlenbeck processes with small Lévy noises

This paper introduces a class of partially observed stochastic processes driven by Ornstein–Uhlenbeck processes with small Lévy noises under continuous-time observations. We discuss the consistency and asymptotic distribution of the trajectory fitting estimators. The numerical simulation results are also presented. By comparing the trajectories and parameter estimation results of the partially observed process with and without small noises, we demonstrate the rationality of incorporating small noise in the partially observed processes.

Optimal subsampling for functional composite quantile regression in massive data

As computer resources become increasingly limited, traditional statistical methods face challenges in analyzing massive data, especially in functional data analysis. To address this issue, subsampling offers a viable solution by significantly reducing computational requirements. This paper introduces a subsampling technique for composite quantile regression, designed for efficient application within the functional linear model on large datasets. We establish the asymptotic distribution of the subsampling estimator and introduce an optimal subsampling method based on the functional L-optimality criterion. Results from simulation studies and the real data analysis consistently demonstrate the superiority of the L-optimality criterion-based optimal subsampling method over the uniform subsampling approach.

Online Monitoring of Structural Change Points Based on Ratio-Type Statistics

For scenarios where the type of structural break in a time series is unknown, this paper proposes a modified ratio-type test statistic to enable effective online monitoring of structural breaks, while circumventing the estimation of long-term variance. Under specific assumptions, we rigorously derive the asymptotic distribution of the test statistic under the null hypothesis and establish its consistency under the alternative hypothesis. In cases where both variance and mean breaks coexist, we introduce a refined mixed-break monitoring procedure based on the consistent estimation of breakpoints. The proposed method first provides consistent estimations of the mean change points and variance change points separately; then, mean and variance removal are performed on original data; finally, the previously removed trend is added back. Compared to traditional monitoring methods, which have to use two test statistics, this method requires only one to simultaneously monitor both types of change points, resulting in a significantly simplified monitoring process. This approach effectively reduces mutual interference between the two types of breaks, thereby enhancing the power of the test. Extensive numerical simulations confirm that this method can accurately detect the presence of structural breaks and reliably identify their types. Finally, case studies are provided to demonstrate the efficacy and practical applicability of the proposed method.

Testing for random interaction: The symmetry assumption

. The F-test for the hypothesis of no interaction effects in any mixed model is valid under the assumption of normality, symmetry, and variance homogeneity of the error terms. We consider the balanced two-way mixed model in which the usual assumptions do not hold. The model allows for dependence of the random main and interaction effects, variance heterogeneity in the random interaction effects and error terms and also do not require normality. We propose the test for testing the hypothesis of no interaction effect in this model. The asymptotic null distribution of the test statistic is the normal distribution and studied under the condition that the number of levels of random effect tends to be large.

Multivariate meta-analysis with a robustified diagonal likelihood function

Multivariate meta-analysis is an efficient tool to analyze multivariate outcomes from independent studies, with the advantage of accounting for correlations between these outcomes. However, existing methods are sensitive to outliers in the data. In this paper, we propose new robust estimation methods for multivariate meta-analysis. In practice, within-study correlations are frequently not reported in studies, conventional robust multivariate methods using modified estimation equations may not be applicable. To address this challenge, we utilize robust functions to create new log-likelihood functions, by only using the diagonal components of the full covariance matrices. This approach bypasses the need for within-study correlations and also avoids the singularity problem of covariance matrices in the computation. Furthermore, the asymptotic distributions can automatically account for the missing correlations between multiple outcomes, enabling valid confidence intervals on functions of parameter estimates. Simulation studies and two real-data analyses are also carried out to demonstrate the advantages of our new robust estimation methods. Our primary focus is on bivariate meta-analysis, although the approaches can be applied more generally.

Unified inference for longitudinal/functional data quantile dynamic additive models

AbstractWe investigate the unified inference of a time‐varying additive model under the quantile regression framework, considering both sparse and dense longitudinal or functional data. For convolution‐type smoothed objective functions, we propose a two‐step method for estimating both the trend and the component functions. Theoretical analysis shows that the two‐step estimators share the same asymptotic distribution as the oracle estimators, while the convergence rates and limiting variance functions differ between sparse and dense situations. However, making a subjective choice between these two cases can lead to incorrect statistical inferences. To address this issue, we develop sandwich formulas for variance estimations. This allows us to establish a unified inference without the need to decide whether the data are sparse or dense. Via simulation studies, we assess the finite‐sample performance of the proposed methods. Finally, analyses of two different types of real data illustrate our proposed methods.

A Grouped GEE Framework for Quantile Regression in Heterogeneous Longitudinal Data Analysis

ABSTRACTIn longitudinal data analysis, identifying subgroups of subjects with heterogeneous parameters is crucial for understanding complex data structures. To address the challenges posed by within‐subject correlation and heteroscedasticity across subjects, we propose a novel grouped generalized estimating equation (GEE) method for quantile regression (QR). Under the assumption that subjects can be partitioned into a finite number of groups, where individuals within the same group share identical regression coefficients at a given quantile level, we develop a computationally efficient three‐step algorithm that simultaneously performs subject grouping, estimates QR coefficients and infers the correlation matrix. Theoretically, we establish the asymptotic distribution of the group‐specific QR coefficient estimator by demonstrating its asymptotic equivalence to the infeasible estimator with known group membership, even when both dimensions of the longitudinal data diverge. Numerical studies and theoretical analysis show that incorporating the estimated correlation matrix significantly improves the efficiency of both coefficient estimation and group identification, which provides an efficient and flexible framework for analysing heterogeneous longitudinal data.

Direct and inverse spectral problems for the Schrödinger operator with double generalized Regge boundary conditions

AbstractIn this paper, we study the direct and inverse spectral problems for the Schrödinger operator with two generalized Regge boundary conditions. For the direct problem, we give the properties of the spectrum, including the asymptotic distribution of the eigenvalues. For the inverse problems, we prove several uniqueness theorems, including the cases: even potential, two‐spectra, as well as the general partial inverse problem.

Asymptotic distribution of the statistical complexity under the multinomial law

Asymptotic distribution of the statistical complexity under the multinomial law