Empirical Process Research Articles

The functional central limit theorem for Markov-switching GARCH model

In this paper we consider the Markov Switching GARCH model suggested by Haas, Mittnik, and Paolella(2004). We show under proper assumptions that the functional central limit theorems hold for the process, the square of the process, and regime variances. The functional central limit theorem for a linear combination of regime variances is also obtained.

On a conjecture of Talagrand on selector processes and a consequence on positive empirical processes

On a conjecture of Talagrand on selector processes and a consequence on positive empirical processes

The Reliability Modeling and Evaluation of a Cusped Field Thruster When Undertaking a Gravitational Wave Detection Mission

The propulsion system, particularly electric propulsion, holds immense significance in the context of gravitational wave detection missions. One of the key factors of a deep space exploration mission is the lifetime of the electric propulsion. Ensuring the high reliability of the propulsion system is of paramount importance; however, achieving this is challenging in the absence of adequate failure data. Conducting ground tests for a thruster tends to encounter two limitations: a lack of failure data and time constraints. To address these challenges, we propose a semi-physics sputtering method that combines a physical erosion model with empirical processes. In this study, we focus on evaluating the lifespan of a cusped field thruster (CFT) for potential application in gravitational wave detection missions. Our analysis revolves around modeling non-conservative forces in a space environment and examining their impact on a thruster’s longevity. The results indicate that, in gravitational wave missions, the survival rate of a thruster’s lifespan at 8000 h is 0.75. At a constant voltage of 500 V, the maximum corrosion depth after 5000 h is 3.1 mm, while the minimum is 0.49 mm.

Functional central limit theorems for rough volatility

AbstractThe non-Markovian nature of rough volatility makes Monte Carlo methods challenging, and it is in fact a major challenge to develop fast and accurate simulation algorithms. We provide an efficient one for stochastic Volterra processes, based on an extension of Donsker’s approximation of Brownian motion to the fractional Brownian case with arbitrary Hurst exponent $H \in (0,1)$ H ∈ ( 0 , 1 ) . Some of the most relevant consequences of this ‘rough Donsker (rDonsker) theorem’ are functional weak convergence results in Skorokhod space for discrete approximations of a large class of rough stochastic volatility models. This justifies the validity of simple and easy-to-implement Monte Carlo methods, for which we provide detailed numerical recipes. We test these against the current benchmark hybrid scheme and find remarkable agreement (for a large range of values of $H$ H ). Our rDonsker theorem further provides a weak convergence proof for the hybrid scheme itself and allows constructing binomial trees for rough volatility models, the first available scheme (in the rough volatility context) for early exercise options such as American or Bermudan options.

Microcrack monitoring and fracture evolution of coal and rock using integrated acoustic emission and digital image correlation techniques.

The mechanical properties of a coal-rock body were examined through uniaxial compression tests, and the rupture process of the coal-rock body was monitored in real time using a combined acoustic emission (AE) monitoring system and a digital image correlation (DIC) full-field strain measurement system. From a comparison of the mechanical properties of coal and sandstone, clear differences are apparent regarding the uniaxial compressive strength, deformation characteristics, and damage mode; the brittle failure characteristics of the coal samples are also more evident. The change in AE energy reflects the accumulation and release of elastic energy during the rupture process, and the evolution of AE localization points under different stress levels can effectively reflect rupture propagation. Further, the DIC full-field strain measurement method can quantitatively monitor the evolution of the displacement and strain fields at the marking point and surface simultaneously, thereby overcoming the limitations of traditional empirical and qualitative rupture processes. During monitoring, the AE focuses on the internal rupture of the specimen and the DIC focuses on the surface deformation. These complement each other and reflect the rupture process more comprehensively.

A uniform kernel trick for high and infinite-dimensional two-sample problems

We use a suitable version of the so-called ”kernel trick” to devise two-sample tests, especially focussed on high-dimensional and functional data. Our proposal entails a simplification of the practical problem of selecting an appropriate kernel function. Specifically, we apply a uniform variant of the kernel trick which involves the supremum within a class of kernel-based distances. We obtain the asymptotic distribution of the test statistic under the null and alternative hypotheses. The proofs rely on empirical processes theory, combined with the delta method and Hadamard directional differentiability techniques, and functional Karhunen-Loève-type expansions of the underlying processes. This methodology has some advantages over other standard approaches in the literature. We also give some experimental insight into the performance of our proposal compared to other kernel-based approaches (the original proposal by Borgwardt et al. (2006) and some variants based on splitting methods) as well as tests based on energy distances (Rizzo et al., 2017).

A semantic segmentation algorithm for automated rapid melt pool identification from cross-sectional micrographs

Melt pool contours are an important empirical process measurement for metallic additive manufacturing. Melt pool boundaries indicate the maximal extent of the melting temperature contour. Melt pool boundaries characterize the effects of process parameters on melt pool size and variability and provide calibration and validation data for process simulations predicting thermal fields and microstructure. Serial cross sectioning of samples can produce numerous micrographs valuable for building robust melt pool contour datasets. However, the acquisition of contours from micrographs by hand is error prone and prohibitively time consuming for datasets large enough to capture variation within individual scan tracks and across a range of process conditions. In this work, we propose a semantic segmentation algorithm for automating melt pool contour acquisition from micrographs. The proposed algorithm is demonstrated on datasets from two different single scan track experiments conducted on a Powder Bed Fusion-Laser Beam platform using a bare Ti-6Al-4 V plate and an Inconel 718 plate with a 40 μm powder layer at different preheat temperatures. The algorithm was successfully trained and predicted the melt pool boundaries in the 300 image Ti-6Al-4 V dataset with 95% accuracy and an estimated reduction of 91% in processing time compared to data acquisition by hand. For the 1201 image Inconel 718 dataset, the algorithm achieved 89% accuracy and an estimated 99% time savings compared to hand measurements. The results for both datasets are demonstrated with a discussion about observed variability in the melt pool characteristics.

The Law of the Iterated Logarithm for Lp-Norms of Kernel Estimators of Cumulative Distribution Functions

In this paper, we consider the strong convergence of Lp-norms (p≥1) of a kernel estimator of a cumulative distribution function (CDF). Under some mild conditions, the law of the iterated logarithm (LIL) for the Lp-norms of empirical processes is extended to the kernel estimator of the CDF.

Detecting Weak Distribution Shifts via Displacement Interpolation

Detecting weak, systematic distribution shifts and quantitatively modeling individual, heterogeneous responses to policies or incentives have found increasing empirical applications in social and economic sciences. Given two probability distributions P (null) and Q (alternative), we study the problem of detecting weak distribution shift deviating from the null P toward the alternative Q, where the level of deviation vanishes as a function of n, the sample size. We propose a model for weak distribution shifts via displacement interpolation between P and Q, drawing from the optimal transport theory. We study a hypothesis testing procedure based on the Wasserstein distance, derive sharp conditions under which detection is possible, and provide the exact characterization of the asymptotic Type I and Type II errors at the detection boundary using empirical processes. We demonstrate how the proposed testing procedure works in modeling and detecting weak distribution shifts in real data sets using two empirical examples: distribution shifts in consumer spending after COVID-19, and heterogeneity in the published p-values of statistical tests in journals across different disciplines.

Neural Network Approximation for Pessimistic Offline Reinforcement Learning

Deep reinforcement learning (RL) has shown remarkable success in specific offline decision-making scenarios, yet its theoretical guarantees are still under development. Existing works on offline RL theory primarily emphasize a few trivial settings, such as linear MDP or general function approximation with strong assumptions and independent data, which lack guidance for practical use. The coupling of deep learning and Bellman residuals makes this problem challenging, in addition to the difficulty of data dependence. In this paper, we establish a non-asymptotic estimation error of pessimistic offline RL using general neural network approximation with C-mixing data regarding the structure of networks, the dimension of datasets, and the concentrability of data coverage, under mild assumptions. Our result shows that the estimation error consists of two parts: the first converges to zero at a desired rate on the sample size with partially controllable concentrability, and the second becomes negligible if the residual constraint is tight. This result demonstrates the explicit efficiency of deep adversarial offline RL frameworks. We utilize the empirical process tool for C-mixing sequences and the neural network approximation theory for the Holder class to achieve this. We also develop methods to bound the Bellman estimation error caused by function approximation with empirical Bellman constraint perturbations. Additionally, we present a result that lessens the curse of dimensionality using data with low intrinsic dimensionality and function classes with low complexity. Our estimation provides valuable insights into the development of deep offline RL and guidance for algorithm model design.

Robust estimation of a regression function in exponential families

We observe n pairs of independent (but not necessarily i.i.d.) random variables X1=(W1,Y1),…,Xn=(Wn,Yn) and tackle the problem of estimating the conditional distributions Qi⋆(wi) of Yi given Wi=wi for all i∈{1,…,n}. Even though these might not be true, we base our estimator on the assumptions that the data are i.i.d. and the conditional distributions of Yi given Wi=wi belong to a one parameter exponential family Q¯ with parameter space given by an interval I. More precisely, we pretend that these conditional distributions take the form Qθ(wi)∈Q¯ for some θ that belongs to a VC-class Θ¯ of functions with values in I. For each i∈{1,…,n}, we estimate Qi⋆(wi) by a distribution of the same form, i.e. Qθ̂(wi)∈Q¯, where θ̂=θ̂(X1,…,Xn) is a well-chosen estimator with values in Θ¯. We establish non-asymptotic exponential inequalities for the upper deviations of a Hellinger-type distance between the true conditional distributions of the data and the estimated one based on the exponential family Q¯ and the class of functions Θ¯ we chose. We show that our estimation strategy is robust to model misspecification, contamination and the presence of outliers. Besides, when the data are truly i.i.d., the exponential family Q¯ is suitably parametrized and the conditional distributions Qi⋆(wi) of the form Qθ⋆(wi)∈Q¯ for some unknown Hölderian function θ⋆ with values in I, we prove that the estimator θ̂ of θ⋆ is minimax (up to a logarithmic factor). Finally, we provide an algorithm for calculating θ̂ when Θ¯ is a VC-class of functions of low or moderate dimension and we carry out a simulation study to compare its performance to that of the MLE and median-based estimators. The proof of our main result relies on an upper bound, with explicit numerical constants, on the expectation of the supremum of an empirical process over a VC-subgraph class. This bound can be of independent interest.

Analysis of the manufacturing of concrete paving blocks with by-products from thermoelectric power plant

The key factor in constructing urban roads applying interlocking concrete block pavement (ICBP) is to employ an appropriate design method for better effectiveness. Concrete paving blocks (CPB) can also be manufactured using industrial by-products. Then, this paper aims to analyze CPB with coal bottom ash (CBA) from a Thermoelectric Power Plant through mechanical characterization and design methods for ICBP. To this end, tests, such as void index, water absorption, compressive strength, and elasticity modulus, were done on CPB for each mixture. Furthermore, empirical and mechanistic-empirical design methods for ICBP were applied. Then, four dry concrete mixtures (two with CBA and two without CBA) were designed with two water/cement ratios (0.63 and 0.73) to produce rectangular CPB measuring 20 cm × 10 cm × 8 cm (length × width × thickness) by a vibro-press machine. Furthermore, ICBP sections were designed, applying six simulations from empirical and mechanistic-empirical design methods. The results showed that the mixtures A73 and B73 with more water, i.e., water/cement ratio of 0.73, presented higher characteristic compressive strength at 28 days (25.34 MPa and 15.88 MPa, respectively) than the other mixes A63 and B63 with less water, i.e., water/cement ratio of 0.63 (these two mixes showed 15.85 MPa). Furthermore, the CPB application for bike paths or parking areas was allowed according to the Australian specification of 15 MPa for compressive strength test. Also, all ICBP sections from the mechanistic-empirical design required approximately 24.9% fewer materials for the sub-base layer than the empirical process. Finally, the CPB application is possible for areas with light traffic, and the ICBP technology was more feasible with the mechanistic-empirical design method.

Overlapping Batch Confidence Intervals on Statistical Functionals Constructed from Time Series: Application to Quantiles, Optimization, and Estimation

We propose a general purpose confidence interval procedure (CIP) for statistical functionals constructed using data from a stationary time series. The procedures we propose are based on derived distribution-free analogues of the χ 2 and Student’s t random variables for the statistical functional context, and hence apply in a wide variety of settings including quantile estimation, gradient estimation, M-estimation, CVAR-estimation, and arrival process rate estimation, apart from more traditional statistical settings. Like the method of subsampling, we use overlapping batches of time series data to estimate the underlying variance parameter; unlike subsampling and the bootstrap, however, we assume that the implied point estimator of the statistical functional obeys a central limit theorem (CLT) to help identify the weak asymptotics (called OB-x limits, x=I,II,III) of batched Studentized statistics. The OB-x limits, certain functionals of the Wiener process parameterized by the size of the batches and the extent of their overlap, form the essential machinery for characterizing dependence, and consequently the correctness of the proposed CIPs. The message from extensive numerical experimentation is that in settings where a functional CLT on the point estimator is in effect, using large overlapping batches alongside OB-x critical values yields confidence intervals that are often of significantly higher quality than those obtained from more generic methods like subsampling or the bootstrap. We illustrate using examples from CVaR estimation, ARMA parameter estimation, and NHPP rate estimation; R and MATLAB code for OB-x critical values is available at  web.ics.purdue.edu/ ∼ pasupath .

Limit theorems for a class of processes generalizing the U-empirical process

In this paper, we develop theory and tools for studying U-processes, a natural higher-order generalization of the empirical processes. We introduce a class of random discrete U-measures that generalize the empirical U-measure. We establish a Glivenko-Cantelli and a Donsker theorem under conditions on entropy numbers prevalent in the theory of empirical processes. These results are proved under some standard structural conditions on the Vapnik-Chervonenkis classes of functions and some mild conditions on the model. The uniform limit theorems discussed in this paper are key tools for many further developments involving empirical process techniques. Our results are applied to prove the asymptotic normality of Liu's simplicial median. We conclude this paper by extending Anscombe's central limit theorem to encompass randomly stopped U-processes, building upon its application to randomly stopped sums of independent random variables.

Semiparametric Probit Regression Model with General Interval-Censored Failure Time Data

Interval-censored data frequently arise in various biomedical areas involving periodical follow-ups where the failure or event time of interest cannot be observed exactly but is only known to fall into a time interval. This article considers a semiparametric probit regression model, a valuable alternative to other commonly used semiparametric models in survival analysis, to investigate potential risk factors for the interval-censored failure time of interest. We develop an expectation-maximization (EM) algorithm to conduct the pseudo maximum likelihood estimation (MLE) using the working independence strategy for general or mixed-case interval-censored data. The resulting estimators of regression parameters are shown to be consistent and asymptotically normal with the empirical process techniques. In addition, we propose a novel penalized EM algorithm for simultaneously achieving variable selection and parameter estimation in the case of high-dimensional covariates. The proposed variable selection method can be readily implemented with some existing software and considerably reduces the estimation error of the proposed pseudo-MLE approach. Simulation studies demonstrate the satisfactory performance of the proposed methods. An application to a set of interval-censored data on prostate cancer further confirms the utility of the methodology. Supplementary materials for this article are available online.

Functional Uniform-in-Bandwidth Moderate Deviation Principle for the Local Empirical Processes Involving Functional Data

Functional Uniform-in-Bandwidth Moderate Deviation Principle for the Local Empirical Processes Involving Functional Data

Critical Thinking Skills in Physics Learning of Students at SMAN 8 Makassar

This research aims to analyze critical thinking skills in students' physics learning at SMAN 8 Makassar. It is descriptive research employing a survey method. The research subjects are 271 students from the 10th grade at SMAN 8 Makassar, distributed across nine classes. Students' critical thinking skills are measured using a multiple-choice test instrument with indicators such as interpretation, analysis, and inference. The data collection technique used is a critical thinking skills test consisting of multiple-choice questions covering critical thinking indicators. The researcher developed these questions and underwent validation by experts and empirical validation processes. The expert validation results show that the test instrument is valid with a score above 0.75. The research results are analyzed using descriptive statistical analysis, indicating that students' critical thinking skills are still in the low category. The critical thinking skills level for interpretation is 29.97%, for analysis is 31.30%, and for inference is 38.73%. This implies that teachers must strive for innovative teaching methods to empower students' critical thinking skills. The implications of this research are expected to provide support and opportunities for teachers to improve the quality of the learning process, enabling students to develop critical thinking skills in solving physics problems and producing high-achieving students.

Data-driven sensor delay estimation in industrial processes using multivariate projection methods

A key step in preparing industrial data for multivariate statistical modelling of (batch-)continuous processes is the estimation of sensor delays along the production line, to use as a correction for understanding relationships between them. Without such a correction, the measurements collected from the sensors do not all relate to the same portion of material travelling through the plant, which is where from a physical point of view the correlations are expected to arise from. Most methods to do these corrections currently reported in literature are limited in that they optimize only correlations bivariately between the different sensors, despite the eventual goal of being understood as a multivariate process model. This work proposes several methods for multivariate sensor delay estimation, based on multivariate projection methods, and critically investigates these using simulated and empirical process data from two industrial facilities. The objective is to provide more clarity on if, how, and why taking into account these multivariate correlations using such projection methods is beneficial. One of the proposed methods, named PLS-SEQ, was shown to outperform bivariate sensor delay estimation, which is regarded as benchmark method in this work, for all datasets. The average reduction in delay estimation error achieved by using PLS-SEQ is between 6% and 64%. The other newly proposed multivariate projection-based methods did not show such unanimous improvement in accuracy, but helped identifying the presence of sensor clusters in industrial data as a novel key challenge for sensor delay estimation. The presented results illustrate that projection-based multivariate methods, in particular the newly proposed method PLS-SEQ, show promising potential for sensor delay estimation, outperforming the bivariate sensor delay estimation methods used as benchmark. Ultimately, using these methods can improve the predictive and descriptive quality of statistical models for key industrial processes.

COINTEGRATING POLYNOMIAL REGRESSIONS: ROBUSTNESS OF FULLY MODIFIED OLS

Cointegrating polynomial regressions (CPRs) include deterministic variables, integrated variables, and their powers as explanatory variables. Based on a novel kernel-weighted limit result and a novel functional central limit theorem, this paper shows that the fully modified ordinary least squares (FM-OLS) estimator of Phillips and Hansen (1990, Review of Economic Studies 57, 99–125) is robust to being used in CPRs. Being used in CPRs refers to a widespread empirical practice that treats the integrated variables and their powers, incorrectly, as a vector of integrated variables and uses textbook FM-OLS. Robustness means that this “formal” FM-OLS practice leads to a zero mean Gaussian mixture limiting distribution that coincides with the limiting distribution of the Wagner and Hong (2016, Econometric Theory 32, 1289–1315) application of the FM estimation principle to the CPR case. The only restriction for this result to hold is that all integrated variables to power one are included as regressors. Even though simulation results indicate performance advantages of the Wagner and Hong (2016, Econometric Theory 32, 1289–1315) estimator, partly even in large samples, the results of the paper give an asymptotic foundation to “formal” FM-OLS and thus enlarge the usability of the Phillips and Hansen (1990, Review of Economic Studies 57, 99–125) estimator implemented in many software packages.

Regression analysis of group-tested current status data

Summary Group testing is an effective way to reduce the time and cost associated with conducting large-scale screening for infectious diseases. Benefits are realized through testing pools formed by combining specimens, such as blood or urine, from different individuals. In some studies, individuals are assessed only once and a time-to-event endpoint is recorded, for example, the time until infection. Combining group testing with this type of endpoint results in group-tested current status data (Petito & Jewell, 2016). To analyse these complex data, we propose methods that estimate a proportional hazard regression model based on test outcomes from measuring the pools. A sieve maximum likelihood estimation approach is developed that approximates the cumulative baseline hazard function with a piecewise constant function. To identify the sieve estimator, a computationally efficient expectation-maximization algorithm is derived by using data augmentation. Asymptotic properties of both the parametric and nonparametric components of the sieve estimator are then established by applying modern empirical process theory. Numerical results from simulation studies show that our proposed method performs nominally and has advantages over the corresponding estimation method based on individual testing results. We illustrate our work by analysing a chlamydia dataset collected by the State Hygienic Laboratory at the University of Iowa.