Dependent Random Vectors Research Articles

Polynomial-chaos-based conditional statistics for probabilistic learning with heterogeneous data applied to atomic collisions of Helium on graphite substrate

A formulation and an algorithm are presented to construct a truncated polynomial chaos representation of a vector-valued random output. This representation depends on a vector-valued random input with a known probability measure and a vector-valued random latent variable with an unknown probability measure. The construction of this PCE representation relies solely on a training set comprising a small number of independent realizations of the non-Gaussian dependent random output and input vectors. The training set consists of heterogeneous data, which poses challenges in accurately estimating the chaos coefficients. Despite the heterogeneity of the data, the proposed formulation and algorithm allow for the construction of a highly accurate global surrogate model. Additionally, we propose an alternative approach by constructing a surrogate model based on prior separation of the heterogeneous dataset into subsets, each containing “quasi-homogeneous” data. The separation method is designed to account for a partial overlap of the probability measure supports associated with the subsets. The identification of the PCE is performed offline. By utilizing the PCE, a fast online surrogate model is obtained, enabling analysis of large dynamical systems beyond the computational capabilities currently available. An application to atomic collisions of Helium on a graphite substrate is presented, where the training set was generated by Molecular Dynamics simulations done in a previous paper. The obtained results demonstrate accuracy of the proposed approach.

The complete moment convergence for coordinatewise pairwise negatively quadrant dependent random vectors in Hilbert space

In this article, we investigate the complete moment convergence for a sequence of coordinatewise pairwise negatively quadrant dependent random vectors in Hilbert space, which is coordinatewise weakly upper bounded by a random vector under special case.

On the Strong Laws of Large Numbers for Sequences of Blockwise Pairwise and Coordinatewise Negatively Dependent Random Vectors in Hilbert Spaces

On the Strong Laws of Large Numbers for Sequences of Blockwise Pairwise and Coordinatewise Negatively Dependent Random Vectors in Hilbert Spaces

On complete convergence for weighted sums of pairwise and coordinatewise negatively dependent random vectors in Hilbert spaces

In this paper, we establish the complete convergence theorem and the Marcinkiewicz–Zymund type strong law of large numbers for sequences of pairwise and coordinatewise negatively dependent random vectors fXn; n ≥ 1g taking values in Hilbert spaces

High-dimensional central limit theorems by Stein’s method

We obtain explicit error bounds for the d-dimensional normal approximation on hyperrectangles for a random vector that has a Stein kernel, or admits an exchangeable pair coupling, or is a nonlinear statistic of independent random variables or a sum of n locally dependent random vectors. We assume the approximating normal distribution has a nonsingular covariance matrix. The error bounds vanish even when the dimension d is much larger than the sample size n. We prove our main results using the approach of Götze (1991) in Stein’s method, together with modifications of an estimate of Anderson, Hall and Titterington (1998) and a smoothing inequality of Bhattacharya and Rao (1976). For sums of n independent and identically distributed isotropic random vectors having a log-concave density, we obtain an error bound that is optimal up to a logn factor. We also discuss an application to multiple Wiener–Itô integrals.

Weak Laws of Large Numbers for Negatively Superadditive Dependent Random Vectors in Hilbert Spaces

Let $\{X_{n}, {n}\in \mathbb{N}\}$ be a sequence of negatively superadditive dependent random vectors taking values in a real separable Hilbert space. In this paper, we present the weak laws of large numbers for weighted sums (with or without random indices) of $\{X_{n}, {n}\in \mathbb{N}\}$.

Central Limit Theorems for Weighted Sums of Dependent Random Vectors in Hilbert Spaces via the Theory of the Regular Variation

In this paper, based on the theory of regularly varying functions we study central limit theorems for the weighted sum $$S_n=\sum _{j=1}^{m_n}c_{nj}X_{nj}$$ , where $$(X_{nj};1\le j \le m_n,n\ge 1)$$ is a Hilbert-space-valued identically distributed martingale difference array and $$(c_{nj};1\le j \le m_n,n\ge 1)$$ is an array of real numbers. As an application, we present a central limit theorem for moving average processes of martingale differences.

Dispersive order comparisons on extreme order statistics from homogeneous dependent random vectors

Abstract In this paper, we investigate sufficient conditions for preservation property of the dispersive order for the smallest and largest order statistics of homogeneous dependent random vectors. Moreover, we establish sufficient conditions for ordering with the dispersive order the largest order statistics from dependent homogeneous samples of different sizes.

Limiting Behavior of the Partial Sum for Negatively Superadditive Dependent Random Vectors in Hilbert Space

In this paper, We study the complete convergence and Lp- convergence for the maximum of the partial sum of negatively superadditive dependent random vectors in Hilbert space. The results extend the corresponding ones of Ko (Ko, 2020) to H-valued negatively superadditive dependent random vectors.

Vehicle Routing with Space- and Time-Correlated Stochastic Travel Times: Evaluating the Objective Function

We study how to model and handle correlated travel times in two-stage stochastic vehicle-routing problems. We allow these travel times to be correlated in time and space; that is, the travel time on one link in one period can be correlated to travel times on the same link in the next and previous periods as well as travel times on neighboring links (links sharing a node) in both the same and the following periods. Hence, we are handling a very high-dimensional dependent random vector. We discuss how such vehicle-routing problems should be modeled in time and space, how the random vector can be represented, and how scenarios (discretizations) can meaningfully be generated to be used in a stochastic program. We assume that the stochastic vehicle-routing problem is being solved by a search heuristic and focus on the objective function evaluation for any given solution. Numerical procedures are given and tested. As an example, our largest case has 142 nodes, 418 road links, and 60 time periods, leading to 25,080 dependent random variables. To achieve an objective function evaluation stability of 1%, we need only 15 scenarios for problem instances with 64 customer nodes and nine vehicles.

Convergence properties for weighted sums of weakly dependent random vectors in Hilbert spaces

In this paper, some results on the complete moment convergence, complete convergence, complete integral convergence, and strong law of large numbers for weighted sums of weakly dependent random vectors in Hilbert spaces are established. The results obtained in this paper improve and extend the corresponding ones of Sung [Complete convergence for weighted sums of -mixing random variables, Discrete Dyn. Nat. Soc. 2010 (2010), Article ID 630608, 13 pages], Huan [Baum-Katz type theorems for coordinatewise negatively associated random vectors in Hilbert spaces, Acta Math. Hungar. 144(1) (2014), pp. 132–149], Huan [On complete convergence for sequences of random vectors in Hilbert spaces, Acta Math. Hungar. 147(1) (2015), pp. 205–219], and Ko [The complete moment convergence for CNA random vectors in Hilbert spaces, J. Inequal. Appl. 2017 (2017), Article ID 290, 11 pages].

On the almost sure convergence for sums of negatively superadditive dependent random vectors in Hilbert spaces and its application

This paper develops almost sure convergence for sums of negatively superadditive dependent random vectors in Hilbert spaces, we obtain Chung type SLLN and the Jaite type SLLN for sequences of negatively superadditive dependent random vectors in Hilbert spaces. Rate of convergence is studied through considering almost sure convergence to 0 of tail series. As an application, the almost sure convergence of degenerate von Mises-statistics is investigated.

English

This paper introduces the notion of pairwise and coordinatewise negative dependence for random vectors in Hilbert spaces. Besides giving some classical inequalities, almost sure convergence and complete convergence theorems are established. Some limit theorems are extended to pairwise and coordinatewise negatively dependent random vectors taking values in Hilbert spaces. An illustrative example is also provided.

Uniform Nonparametric Series Inference for Dependent Data with an Application to the Search and Matching Model

This paper concerns the uniform inference for nonparametric series estimators in time-series applications. We develop a strong approximation theory of sample averages of serially dependent random vectors with dimensions growing with the sample size. The strong approximation is first proved for heterogeneous martingale difference arrays and then extended to general mixingales via martingale approximation, readily accommodating a majority of applications in applied econometrics. We use these results to justify the asymptotic validity of a uniform confidence band for series estimators and show that it can also be used to conduct nonparametric specification test for conditional moment restrictions. The validity of high-dimensional heteroskedasticity and autocorrelation consistent (HAC) estimators is established for making feasible inference. The proposed method is broadly useful for forecast evaluation, empirical microstructure, dynamic stochastic equilibrium models and inference problems based on inter-section bounds. We demonstrate the empirical relevance of the proposed method by studying the Mortensen–Pissarides search and matching model for equilibrium unemployment, and shed new light on the unemployment volatility puzzle from an econometric perspective.

Application of Diagnostic Indicators for Biological Test Validation in Assisted Reproductive Technology

In the field of medical reproductive research, the selection of embryos with the best potential for implantation is the main challenge for biologists. Several studies suggest that genes involved in the Oocyte-Cumulus Cell crosstalk could represent candidate genebiomarkers for selecting embryos with the highest implantation potential.Therefore, the principle objective of this study is to verify the transcriptomic experimental data from 21 biomarker genes by RT-qPCR (Real-Time quantitative Polymerase Chain Reaction) of 102 embryonic/cumulus cell samples from patients undergoing in vitro fertilization. Since variability (noises) from various sources (biological, technical, etc.) was observed,there is a reasonable doubt about the capability of these transcriptomic data to provide a reliable and robust pregnancy predictive model. So, our goal is to verify if the genomic signature could be used as biomarker. If so, one can stipulate that the transcriptome is predictable and could generate a reliable mathematical model. Stochastic modelling is based on the Multiple Logistic Regression (MLR) which is bimodal and therefore binary, seems adequate to give a conclusion regarding this genomic signature capacity to predict the absence or presence of Pregnancy (Pr) event. In this work, the observed event will be represented by a dependent random vector Y that takes the value 1 if pregnancy occurs and 0 if not. The prediction value of this vector also depends on the noise (e) induced by the variabilities mentioned above. Bio-stochastic tools such as the ROC (Receiver Operating Characteristic measure) curve and its AUC (Area Under the ROC Curve), the probabilistic likelihood indicators, the Odds Ratio (OR), and finally the Youden Index (YI), appear as a simple and an effective biological decision tools to verify the validity of this genomic signature as biomarker to predict pregnancy (Pr). The analysis of the bio-statistical indicator results indicate that the obtained predictive model is non-discriminant, suggesting a bias in the transcriptomic data.

Random function model for dependent random variables

To characterize the dependency between basic random variables is of paramount importance in engineering practice. However, it is usually difficult to deal with joint probability density function of dependent variates directly. In the present paper, a random function model is proposed for the probabilistic description of 2-dimensional dependent random variables. This random function converts a 2-dimensional dependent random vector into an independent random vector. The undetermined functions in the random function model are found to be the conditional mean and conditional standard deviation function, which could be further specified by observed data. Specifically, in the present paper it is suggested that the shape of the undetermined function, i.e., the conditional mean and standard deviation function in this case, be extracted based on the insight into the embedded physical mechanism, and then the undetermined parameters be identified from observed data. The procedure is illustrated in detail by adopting the relationship between the modulus of elasticity and compressive strength of concrete as an example. The one-dimensional damage evolution mechanism is firstly introduced, yielding the lower limit of the data. Then the viscoelasticity mechanism is advocated to determine the shapes of conditional mean and conditional standard deviation function. The parameters of the model are consequently identified from experimental data. Thereby, a pragmatic model, which could character the dependency between the modulus of elasticity and compressive strength, is proposed for engineering purposes. The comparison to the Copula model shows that the proposed model could capture the probabilistic characteristics of observed data. It is noted that in the proposed model the random function is weakly nonlinear and thus will not worsen the well-posedness of the problem. Besides, the direct dealing with joint probability density function is avoided. The proposed approach could be extended to the probabilistic description of more non-independent random variables.

On mutual information for elliptical distributions: a case of nonlinear dependence of ‘n’ vectors

In this paper, we modeled dependent categorical data via mutual information concept to obtain the measure of statistical dependence. We first derive the entropy and mutual information index for exponential power distribution. These concepts are important and were developed by Shannon in the context of information theory. Several literatures are already published in the case of the multivariate normal distribution. Then we extend these tools to the special case of a full symmetric of multivariate elliptical distributions. The upper bound for the entropy which is attained for the normal density is established. We further derived the nonlinear joint model for dependent random vectors that spans an elliptical vector space to enhance multivariate relationships among non-empty subsets of vectors via multivariate mutual information; based on the assumption that the subsets of each vector and their interactions can be represented in discrete form. To illustrate its application, the multivariate dependency among various sites based on dominance of some attributes were investigated.

A rate optimal procedure for recovering sparse differences between high-dimensional means under dependence

The paper considers the problem of recovering the sparse different components between two high-dimensional means of column-wise dependent random vectors. We show that dependence can be utilized to lower the identification boundary for signal recovery. Moreover, an optimal convergence rate for the marginal false nondiscovery rate (mFNR) is established under dependence. The convergence rate is faster than the optimal rate without dependence. To recover the sparse signal bearing dimensions, we propose a Dependence-Assisted Thresholding and Excising (DATE) procedure, which is shown to be rate optimal for the mFNR with the marginal false discovery rate (mFDR) controlled at a pre-specified level. Extensions of the DATE to recover the differences in contrasts among multiple population means and differences between two covariance matrices are also provided. Simulation studies and case study are given to demonstrate the performance of the proposed signal identification procedure.

Optimal multivariate quota-share reinsurance: A nonparametric mean-CVaR framework

In this paper, the Conditional Value-at-Risk (CVaR) is adopted to measure the total loss of multiple lines of insurance business and two nonparametric estimation methods are introduced to explore the optimal multivariate quota-share reinsurance under a mean-CVaR framework. While almost all the existing literature on optimal reinsurance are based on a probabilistic derivation, the present paper relies on a statistical analysis. The proposed optimal reinsurance models are directly formulated on empirical data and no explicit distributional assumption on the underlying risk vector is required. The resulting nonparametric reinsurance models are convex and computationally amenable, circumventing the difficulty of computing CVaR of the sum of a generally dependent random vector. Statistical consistency of the resulting estimators for the best CVaR is established for both nonparametric models, allowing empirical data to be generated from any stationary process satisfying strong mixing conditions. Finally, numerical experiments are presented to show that a routine bootstrap procedure can capture the distributions of the resulting risk measures well for independent data.

Darling-Erdös-type test for change detection in parameters and variance for stationary VAR models

ABSTRACTThe aim of this article is the construction of the test statistic for the detection of changes in vector autoregressive (AR) models where both AR parameters and the variance matrix of the error term are the subjects of a change. The approximating distribution of the proposed statistic is the Gumbel distribution. The proof stands on the approximation of weakly dependent random vectors by independent ones and by application of Horváth’s extension of Darling-Erdös extremal result for random vectors, see Darling and Erdös (1956) and Horváth (1993). The test statistic is a modification of the likelihood ratio.