Random Variables Research Articles

Quantile-based cumulative Kullback-Leibler divergence in past lifetime: Some properties and applications

This article introduces a quantile-based cumulative Kullback-Leibler (KL) divergence measure for past lifetimes (QPCKL divergence) and studies its various properties. Unlike the cumulative KL divergence using the distribution function, the proposed measure offers an alternative method for computing the divergence between two random variables, especially when one lacks a closed-form distribution function but possesses a closed-form quantile function. A characterization property of a distribution is derived using the constant QPCKL divergence model. Subsequently, a comparison of some important reliability characteristics using the proposed measure is discussed. The QPCKL divergence for a general family of transformation models is derived. We also propose a non parametric estimator for the measure, and simulation studies are carried out for validation. Finally, the importance of the estimator is also illustrated through real data sets useful in survival studies.

Regularity properties of random wavelet series

We study the regularity properties of random wavelet series constructed by multiplying the coefficients of a deterministic wavelet series with unbounded I.I.D. random variables. In particular, we show that, at the opposite to what happens for Fourier series, the randomization of almost every continuous function gives an almost surely nowhere locally bounded function.

Explicit constants in the nonuniform local limit theorem for Poisson binomial random variables

In a recent paper the authors proved a nonuniform local limit theorem concerning normal approximation of the point probabilities P(S=k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$P(S=k)$\\end{document} when S=∑i=1nXi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$S=\\sum_{i=1}^{n}X_{i}$\\end{document} and X1,X2,…,Xn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X_{1},X_{2},\\ldots ,X_{n}$\\end{document} are independent Bernoulli random variables that may have different success probabilities. However, their main result contained an undetermined constant, somewhat limiting its applicability. In this paper we give a nonuniform bound in the same setting but with explicit constants. Our proof uses Stein’s method and, in particular, the K-function and concentration inequality approaches. We also prove a new uniform local limit theorem for Poisson binomial random variables that is used to help simplify the proof in the nonuniform case.

Resilience-based post-disaster repair strategy for integrated public transit networks

The resilience of public transit systems has become a critical issue due to their essential role in urban mobility. However, the challenges of repair selection and sequencing in integrated public transportation networks (IPTN) have not been fully explored. Therefore, we propose a bi-level resilience-based repair strategy optimisation (RRSO) model to investigate the optimisation of resilience in IPTN. To consider trip selection and account for cascade failure processes, a lower-level traffic assignment model was integrated into a single-layer RRSO model. In the traffic assignment model, perceived travel time reflects the traffic impedance of public transportation. Additionally, to assess the performance of the RRSO model under emergency repair conditions, random variables were introduced to represent uncertain factors in the traffic environment. Finally, the results demonstrate that the RRSO model outperforms standard empirical repair strategies. This study provides theoretical insights into the management of urban public transportation.

Simulating non-stationary and non-Gaussian cross-correlated fields using multivariate Karhunen–Loève expansion and L-moments-based Hermite polynomial model

In practical engineering, cross-correlated random fields are often used to model structural materials or random loads containing multiple correlations. Effective and accurate simulation of these cross-correlation fields is an important prerequisite for subsequent reliability analysis and uncertainty quantification of complex systems. Therefore, this paper proposes a new simulation method for non-stationary non-Gaussian cross-correlated random fields to satisfy realistic engineering requirements. In this new method, L-moments-based Hermite polynomials model (LHPM) is extended to non-stationary non-Gaussian cross-correlated fields. Then, a transformation model from non-stationary non-Gaussian correlation matrix function (CMF) to the underlying Gaussian CMF is explicitly given. Multivariate K-L expansion is further used to approximate the underlying Gaussian cross-correlated fields, where the eigenvalue and eigenfunctions are estimated via the Nyström method with uniform weights. The approximation permits the application of few random variables to characterize the entire Gaussian cross-correlated fields. Ultimately, the simulated underlying Gaussian cross-correlated fields are mapped back to the target non-stationary non-Gaussian cross-correlated fields based on LHPM. Three typical examples, including exponential kernel, Wiener process kernel and spatially varying non-Gaussian and nonstationary seismic ground motions, are used to validate the effectiveness of the proposed method. The source code is readily available at: https://github.com/zhaozhao23/Simulation-of-non-stationary-and-non-Gaussian-cross-correlated-fields.

Characterization of the least squares estimator: Mis-specified multivariate isotonic regression model with dependent errors

This article investigates some nice properties of the least squares estimator of multivariate isotonic regression function (denoted as LSEMIR), when the model is mis-specified, and the errors are β \beta -mixing stationary random variables. Under mild conditions, it is observed that the least squares estimator converges uniformly to a certain monotone function, which is closest to the original function in an appropriate sense.

Some New Results on Stochastic Comparisons of Spacings of Generalized Order Statistics from One and Two Samples

Generalized order statistics (GOSs) are often adopted as a tool for providing a unified approach to several stochastic models dealing with ordered random variables. In this contribution, we first recall various useful results based on the notion of total positivity. Then, some stochastic comparisons between spacings of GOSs from one sample, as well as two samples, are developed under the more general assumptions on the parameters of the model. Specifically, the given results deal with the likelihood ratio order, the hazard rate order and the mean residual life order. Finally, an application is demonstrated for sequential systems.

Meixner random variables and their quantum operators

Meixner random variables and their quantum operators

Almost Sure Central Limit Theorem for Error Variance Estimator in Pth-Order Nonlinear Autoregressive Processes

In this paper, under some suitable assumptions, using the Taylor expansion, Borel–Cantelli lemma and the almost sure central limit theorem for independent random variables, the almost sure central limit theorem for error variance estimator in the pth-order nonlinear autoregressive processes with independent and identical distributed errors was established. Four examples, first-order autoregressive processes, self-exciting threshold autoregressive processes, threshold-exponential AR progresses and multilayer perceptrons progress, are given to verify the results.

Multi-Input data ASsembly for joint Analysis (MIASA): A framework for the joint analysis of disjoint sets of variables.

The joint analysis of two datasets [Formula: see text] and [Formula: see text] that describe the same phenomena (e.g. the cellular state), but measure disjoint sets of variables (e.g. mRNA vs. protein levels) is currently challenging. Traditional methods typically analyze single interaction patterns such as variance or covariance. However, problem-tailored external knowledge may contain multiple different information about the interaction between the measured variables. We introduce MIASA, a holistic framework for the joint analysis of multiple different variables. It consists of assembling multiple different information such as similarity vs. association, expressed in terms of interaction-scores or distances, for subsequent clustering/classification. In addition, our framework includes a novel qualitative Euclidean embedding method (qEE-Transition) which enables using Euclidean-distance/vector-based clustering/classification methods on datasets that have a non-Euclidean-based interaction structure. As an alternative to conventional optimization-based multidimensional scaling methods which are prone to uncertainties, our qEE-Transition generates a new vector representation for each element of the dataset union [Formula: see text] in a common Euclidean space while strictly preserving the original ordering of the assembled interaction-distances. To demonstrate our work, we applied the framework to three types of simulated datasets: samples from families of distributions, samples from correlated random variables, and time-courses of statistical moments for three different types of stochastic two-gene interaction models. We then compared different clustering methods with vs. without the qEE-Transition. For all examples, we found that the qEE-Transition followed by Ward clustering had superior performance compared to non-agglomerative clustering methods but had a varied performance against ultrametric-based agglomerative methods. We also tested the qEE-Transition followed by supervised and unsupervised machine learning methods and found promising results, however, more work is needed for optimal parametrization of these methods. As a future perspective, our framework points to the importance of more developments and validation of distance-distribution models aiming to capture multiple-complex interactions between different variables.

Investigating the Influence of Non-Uniform Characteristics of Layered Foundation on Ground Vibration Using an Efficient 2.5D Random Finite Element Method

High-speed train operation may cause vibration near track facilities and propagate far through the ground, affecting people’s lives, work, and normal use of precision instruments in an urban environment. An efficient numerical method is proposed to calculate the non-uniform ground vibration under a moving high-speed railway load. The theory of stochastic variables is used to describe the soil spatial variability of the non-uniform layered elastic ground, and the coupled 2.5D random finite element method (FEM) is proposed to reduce the computational cost without losing accuracy. Vibration propagation and attenuation of the non-uniform layered ground are investigated and the effect of train speed and soil non-homogeneity are analyzed. Results show that (1) at cross speed and high speed, the homogeneity coefficient of the layered ground has the most important influence on the ground vibration amplitude; (2) the upward acceleration is much larger than the downward acceleration at most speeds, and at cross speed and high speed, the acceleration amplitude decreases with the increase in the homogeneity coefficient; (3) as train speed increases from 60 m/s to 130 m/s, the influencing range of the homogeneity coefficient increases to 10 m from 2 m; and (4) the phenomenon of an in increase in local rebound can be observed in the velocity and acceleration attenuation curve at cross speed when the ground soil has a weaker homogeneity.

Quantification of modeling uncertainty in the Rayleigh damping model

AbstractUnderstanding and accurately characterizing energy dissipation mechanisms in civil structures during earthquakes is an important element of seismic assessment and design. The most commonly used model is attributed to Rayleigh. This paper proposes a systematic approach to quantify the uncertainty associated with Rayleigh's damping model. Bayesian calibration with embedded model error is employed to treat the coefficients of the Rayleigh model as random variables using modal damping ratios. Through a numerical example, we illustrate how this approach works and how the calibrated model can address modeling uncertainty associated with the Rayleigh damping model.

Generalized data thinning using sufficient statistics

Our goal is to develop a general strategy to decompose a random variable X into multiple independent random variables, without sacrificing any information about unknown parameters. A recent paper showed that for some well-known natural exponential families, X can be thinned into independent random variables X ( 1 ) , … , X ( K ) , such that X = ∑ k = 1 K X ( k ) . These independent random variables can then be used for various model validation and inference tasks, including in contexts where traditional sample splitting fails. In this paper, we generalize their procedure by relaxing this summation requirement and simply asking that some known function of the independent random variables exactly reconstruct X. This generalization of the procedure serves two purposes. First, it greatly expands the families of distributions for which thinning can be performed. Second, it unifies sample splitting and data thinning, which on the surface seem to be very different, as applications of the same principle. This shared principle is sufficiency. We use this insight to perform generalized thinning operations for a diverse set of families.

Time varying reliability analysis of corroded gas pipelines using copula and importance sampling

Ensuring the safety of gas pipelines is crucial for the reliable and efficient transportation of natural gas. This study introduces a methodology for evaluating the time-varying reliability of corroded gas pipelines. The proposed approach employs both importance sampling and copula theory to effectively address the uncertainties associated with random variables involved and provide efficient estimates of failure probability. Small leak and burst failure modes are specifically investigated within this study. The methodology first establishes a step-by-step procedure for analysis. Subsequently, the study investigates the failure probability of corroded gas pipelines, taking into account both small leak and burst failures, under scenarios involving single and multiple defects. Furthermore, the study examines the influence of correlation strength and dependence structure among the involved random variables using the copula theory. Additionally, the generation of newly formed defects is thoroughly investigated, along with its impact on failure probability. Numerical examples are provided to evaluate the performance of the proposed methodology, comparing it with the benchmark failure probability estimated through Monte Carlo simulation. The results validate the precision and efficiency of the proposed methodology, while offering practical and insightful suggestions for reliability analysis of real-world gas pipelines.

A general method for the estimation of moments of substitution number in molecular evolution

Abstract The random variable “substitution number” Nt , i.e. the number of mutations that have accumulated in a sequence under neutral evolution during a time t, is a cornerstone of the field of molecular evolution. We show here that a complete solution of the moments of this random variable can be obtained explicitly by recurrence, using a simple matrix shift method. This result leads to simple expressions for both the short and long time limits of the moments that can be computed directly from the substitution matrix used to model the neutral evolution. The method developed here is also used to compute the moments of the complementary variable, Tn , the time it takes to accumulate n mutations. The method we develop here necessitates only elementary operations on the substitution matrix and does not involve spectral decomposition.

Probabilistic Stirling numbers and applications

AbstractWe introduce probabilistic Stirling numbers of the first kind $$s_Y(n,k)$$ s Y ( n , k ) associated with a complex-valued random variable Y satisfying appropriate integrability conditions, thus completing the notion of probabilistic Stirling numbers of the second kind $$S_Y(n,k)$$ S Y ( n , k ) previously considered by the first author. Combinatorial interpretations, recursion formulas, and connections between $$s_Y(n,k)$$ s Y ( n , k ) and $$S_Y(n,k)$$ S Y ( n , k ) are given. We show that such numbers describe a large subset of potential polynomials, on the one hand, and the moments of sums of i. i. d. random variables, on the other, establishing their precise asymptotic behavior without appealing to the central limit theorem. We explicitly compute these numbers when Y has a certain familiar distribution, providing at the same time their combinatorial meaning.

High-dimensional partial correlation coefficients: A survey study of estimation Methods

. The partial correlation coefficient (Pcor) measures the degree of association between two random variables after removing the effect of another set of variables, called the controlling variables. When the dimension of these controlling variables is high, estimating Pcor becomes exceedingly challenging, as conventional computational methods fail to provide efficient estimators. Although several methods have been proposed for estimating Pcor in high-dimensional data, they primarily focus on testing whether its value is zero rather than enhancing estimation efficiency. This article concentrates on improving estimation efficiency and surveys existing methods. Through extensive simulation studies on high-dimensional data, we observe that all current methods consistently underestimate the absolute values of Pcor, with the bias being more pronounced when Pcor is positive. By considering a combination of regression equations, X = Z α + ε and Y = Z β + ζ , the method proposed in this article demonstrates a significant improvement over existing approaches. Stocks data in Hong Kong is employed to demonstrate the effectiveness of these methods.

A Short Note on Generating a Random Sample from Finite Mixture Distributions

Computational statistics is a critical skill for professionals in fields such as data science, statistics, and related disciplines. One essential aspect of computational statistics is the ability to simulate random variables from specified probability distributions. Commonly employed techniques for sampling random variables include the inverse transform method, acceptance–rejection method, and Box–Muller transformation, all of which rely on sampling from the uniform (0,1) distribution. A significant concept in statistics is the finite mixture model, characterized by a convex combination of multiple probability density functions. In this paper, we introduce a modified version of the composition method, a standard approach for sampling finite mixture models. Our modification offers the advantage of relying on sampling from the uniform (0,1) distribution, aligning with prevalent methods in computational statistics. This alignment simplifies teaching computational statistics courses, as well as having other benefits. We offer several examples to illustrate the approach.

Mathematical Model of the Process of Data Transmission over the Radio Channel of Cyber-Physical Systems

This article introduces a refined mathematical model to evaluate the quality of mobile radio channels within cyber-physical systems, employing the topological transformation of stochastic networks. The operation of the radio channel is conceptualized as a stochastic network, enabling the derivation of critical metrics such as an equivalent function, mathematical expectation, variance, and the time distribution function of the implemented processes. The model uses the Gamma distribution for the initial distribution functions of random variables, enhancing its analytical precision. A significant advancement of this study is the development of a comprehensive model that describes the data transmission process through phases of connection establishment, information transmission, and connection maintenance. The innovative aspect of this research lies in applying an equivalent function to a stochastic network that includes a logical “AND” node with gamma-distributed incoming branches. The stochastic network presented in the article, which includes a logical “AND” node, helps to elucidate the mechanism for obtaining an equivalent function for such networks, allowing the application area of the GERT method to be expanded. This methodological enhancement extends the previously limited scope of topological transformation methods, which only applied to exponential distribution models for the timing of branch inputs. By integrating a Gamma distribution, the model simplifies the equivalent function and reduces the computational complexity required to assess the radio channel’s quality, ensuring the accuracy needed for engineering calculations. Moreover, the proposed method requires 25–40% fewer series members than the traditional Taylor series decomposition, while maintaining comparable computational complexity for the typical series members. Furthermore, the maximum absolute error in the calculations is capped at 9 × 10−3, which is well within acceptable limits for engineering purposes. Primarily designed for radio channels in cyber-physical systems, the model’s applicability extends to wireless communications, providing a valuable tool for evaluating channel efficiency and security in the face of increasing cyber threats.

Complete moment convergence for widely orthant dependent random variables

The author studies the complete moment convergence for widely orthant dependent random variables. The obtained result in this paper improves the corresponding one of Chen and Sung [A Spitzer-type law of large numbers for widely orthant dependent random variables, Statist. Probab. Lett. 154 (2019), pp. 108544] by obtaining a much stronger conclusion under the same conditions.