Marginal Moments Research Articles

Generalized bivariate Kummer-beta distribution with marginals defined on the unit interval.

In this paper, a generalized bivariate Kummer-beta distribution is proposed. The name derives from the fact that its particular cases include univariate Kummer-beta distributions. This distribution generalizes a number of existing bivariate beta distributions, including Nadarajah's bivariate distributions, Libby and Novick's bivariate beta distribution and a central bivariate Kummer-beta distribution. Various properties associated with this newly introduced distribution are derived. The derived properties include product moments, marginal densities, marginal moments, conditional densities, conditional moments, Rényi entropy and Shannon entropy. Motivated by possible applications in economics, genetics, hydrology, meteorology, nuclear physics, and reliability, we also derive distributions of the product and the ratio of the components following the proposed distribution. Parameter estimation by maximum likelihood method is discussed by deriving expressions for score functions. Inference based on maximum likelihood estimation supposes that the maximum likelihood estimators have zero bias and zero mean squared errors. A simulation study is performed to check this for finite samples.

Score test for marks in Hawkes processes

AbstractA score statistic for detecting the impact of marks in a linear Hawkes self-exciting point process is proposed, with its asymptotic properties, finite sample performance, power properties using simulation and application to real data presented. A major advantage of the proposed inference procedure is that the Hawkes process can be fitted under the null hypothesis that marks do not impact the intensity process. Hence, for a given record of a point process, the intensity process is estimated once only and then assessed against any number of potential marks without refitting the joint likelihood each time. Marks can be multivariate and serially dependent. The score function for any given set of marks is easily constructed as the covariance of functions of future intensities fits the unmarked process with functions of the marks under assessment. The asymptotic distribution of the score statistic is a Chi-squared distribution, with degrees of freedom equal to the number of parameters required to specify the boost function. Model-based or nonparametric estimation of required features of the mark’s marginal moments and serial dependence can be used. Using sample moments of the marks in the test statistic construction does not impact the size and power properties.

Transportability of model‐based estimands in evidence synthesis

In evidence synthesis, effect modifiers are typically described as variables that induce treatment effect heterogeneity at the individual level, through treatment‐covariate interactions in an outcome model parametrized at such level. As such, effect modification is defined with respect to a conditional measure, but marginal effect estimates are required for population‐level decisions in health technology assessment. For noncollapsible measures, purely prognostic variables that are not determinants of treatment response at the individual level may modify marginal effects, even where there is individual‐level treatment effect homogeneity. With heterogeneity, marginal effects for measures that are not directly collapsible cannot be expressed in terms of marginal covariate moments, and generally depend on the joint distribution of conditional effect measure modifiers and purely prognostic variables. There are implications for recommended practices in evidence synthesis. Unadjusted anchored indirect comparisons can be biased in the absence of individual‐level treatment effect heterogeneity, or when marginal covariate moments are balanced across studies. Covariate adjustment may be necessary to account for cross‐study imbalances in joint covariate distributions involving purely prognostic variables. In the absence of individual patient data for the target, covariate adjustment approaches are inherently limited in their ability to remove bias for measures that are not directly collapsible. Directly collapsible measures would facilitate the transportability of marginal effects between studies by: (1) reducing dependence on model‐based covariate adjustment where there is individual‐level treatment effect homogeneity or marginal covariate moments are balanced; and (2) facilitating the selection of baseline covariates for adjustment where there is individual‐level treatment effect heterogeneity.

Estimation of marginal excess moments for Weibull-type distributions

We consider the estimation of the marginal excess moment (MEM), which is defined for a random vector (X, Y) and a parameter β>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta >0$$\\end{document} as E[(X-QX(1-p))+β|Y>QY(1-p)]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {E}[(X-Q_{X}(1-p))_{+}^{\\beta }|Y> Q_{Y}(1-p)]$$\\end{document} provided E|X|β<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {E}|X|^{\\beta }< \\infty $$\\end{document}, and where y+:=max(0,y)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$y_{+}:=\\max (0,y)$$\\end{document}, QX\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q_{X}$$\\end{document} and QY\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q_{Y}$$\\end{document} are the quantile functions of X and Y respectively, and p∈(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\in (0,1)$$\\end{document}. Our interest is in the situation where the random variable X is of Weibull-type while the distribution of Y is kept general, the extreme dependence structure of (X, Y) converges to that of a bivariate extreme value distribution, and we let p↓0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \\downarrow 0$$\\end{document} as the sample size n→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\rightarrow \\infty $$\\end{document}. By using extreme value arguments we introduce an estimator for the marginal excess moment and we derive its limiting distribution. The finite sample properties of the proposed estimator are evaluated with a simulation study and the practical applicability is illustrated on a dataset of wave heights and wind speeds.

Survival Analysis Without Sharing of Individual Patient Data by Using a Gaussian Copula.

Cox regression and Kaplan-Meier estimations are often needed in clinical research and this requires access to individual patient data (IPD). However, IPD cannot always be shared because of privacy or proprietary restrictions, which complicates the making of such estimations. We propose a method that generates pseudodata replacing the IPD by only sharing non-disclosive aggregates such as IPD marginal moments and a correlation matrix. Such aggregates are collected by a central computer and input as parameters to a Gaussian copula (GC) that generates the pseudodata. Survival inferences are computed on the pseudodata as if it were the IPD. Using practical examples we demonstrate the utility of the method, via the amount of IPD inferential content recoverable by the GC. We compare GC to a summary-based meta-analysis and an IPD bootstrap distributed across several centers. Other pseudodata approaches are also considered. In the empirical results, GC approximates the utility of the IPD bootstrap although it might yield more conservative inferences and it might have limitations in subgroup analyses. Overall, GC avoids many legal problems related to IPD privacy or property while enabling approximation of common IPD survival analyses otherwise difficult to conduct. Sharing more IPD aggregates than is currently practiced could facilitate "second purpose"-research and relax concerns regarding IPD access.

Investigation on the sampling distributions of non-Gaussian wind pressure skewness and kurtosis

Accurate estimation of skewness and kurtosis is crucial for addressing issues related to non-Gaussian wind pressure distribution fitting, signal simulation, and extreme estimation. The skewness and kurtosis determined from finite-length signals are inherent random variables characterized by obvious value fluctuations. Very limited research has been conducted on the sampling distributions of skewness and kurtosis for non-Gaussian wind pressure, motivating the present study. Firstly, the investigation focuses on the non-Gaussian white noise. The expressions of main statistical indicators such as the variance, covariance, correlation coefficient, skewness, and kurtosis of the two sampling distributions are derived and then verified using the Hermite polynomial model. The primary factors influencing the sampling distribution variance (SDV) of white noise are analyzed, including the signal length and marginal moments of the parent distribution. Secondly, the concentration shifts to colored non-Gaussian processes. The composition of SDV is analyzed through theoretical derivation. On this basis, a method for estimating SDV values using the Gaussian process regression model is proposed, with accuracy and feasibility verified based on the long-duration wind pressure data measured in wind tunnel. Furthermore, the relationship between the SDVs of wind pressure processes, signal length, higher-order correlation functions, and marginal moments of the parent process is discussed.

Measure-valued affine and polynomial diffusions

We introduce a class of measure-valued processes, which – in analogy to their finite dimensional counterparts – will be called measure-valued polynomial diffusions. We show the so-called moment formula, i.e. a representation of the conditional marginal moments via a system of finite dimensional linear PDEs. Furthermore, we characterize the corresponding infinitesimal generators obtaining a representation analogous to polynomial diffusions on R+m, in cases where their domain is large enough. In general the infinite dimensional setting allows for richer specifications strictly beyond this representation. As a special case, we recover measure-valued affine diffusions, sometimes also called Dawson–Watanabe superprocesses. From a mathematical finance point of view, the polynomial framework is especially attractive since it allows to transfer many famous finite dimensional models and their tractability properties to an infinite dimensional measure-valued setting.

On the Bivariate Extension of the Extended Standard U-quadratic Distribution

This paper derives the bivariate version of the extended standard U-quadratic (eSU) distribution using the method of compounding or pseudo family of distributions. The joint probability and cumulative distribution functions of the derived distribution are obtained and it is observed that the said distribution can generate bivariate shape with the following properties: $(i)$ $X$ and $Y$ have bathtub shapes; $(ii)$ $X$ has constant and $Y$ has bathtub shapes; and $(iii)$ $X$ has inverted bathtub and $Y$ has bathtub shapes. Moreover, some properties of the derived distribution such as the marginal distribution, conditional distributions, conditional moments, product and ratio moments are derived. Further, the maximum likelihood estimation is performed to estimate the parameters of the derived distribution. Also, a simulation study is carried out to evaluate the behavior of the estimates of the parameters. Moreover, we derive a new bivariate Kumaraswamy distribution and use it to simulate bivariate data with $X$ and $Y$ having bathtub shapes. Furthermore, A new bivariate version of the Cubic Transmuted Uniform (CTU) distribution is also derived. Finally, the proposed Bivariate eSU distribution is applied to simulated data and compared with the said Bivarite Cubic Transmuted Uniform distribution. The result shows that the proposed Bivariate eSU distribution provides a better fit for the said simulated dataset as compared with the Bivariate CTU distribution.

A tri-variate moment projection method for multi-dimensional particle population balance dynamics

This study develops a tri-variate moment projection method (TVMPM) for solving particle dynamics problems governed by the population balance equations involving any three internal coordinates of particles, such as volume, surface area, component mass, etc. By leveraging the concept of conditional moments, the multi-dimensional moments are expressed as the product of marginal moments and several conditional moments, facilitating the solution of high-dimensional moments. To obtain abscissas and conditional weights in different dimensions for reconstructing moments, a three-dimensional (3-D) adaptive Blumstein-Wheeler algorithm is proposed and implemented, which can also improve the stability in the third dimension, where ill-conditioning issues are more likely to arise. More importantly, by fixing the smallest particle abscissas, TVMPM can directly track the weight of the smallest particles, thereby closing the shrinkage and fragmentation terms. An analysis of the sources of errors arising from this approach is also presented. To mitigate potential interference from external factors, constant kernels for inception, growth, coagulation, shrinkage and fragmentation are employed to validate the effectiveness of TVMPM. The resulting moments from TVMPM are computed for both individual and combined processes, and subsequently compared with the moments derived from the direct simulation algorithm (DSA). The results demonstrate that TVMPM maintains a high level of accuracy across various numbers of quadrature nodes for particle dynamics with 3-D internal coordinates, while significantly reducing computational efforts compared to DSA. This study reveals that the developed algorithms and framework are promising for further extending MPM to higher dimensions. Moreover, owing to its accuracy and efficiency, TVMPM shows great potential for implementation in computational fluid dynamics (CFD) for particle dynamics in realistic systems.

A New Type of Bivariate-Gamma Probability Function

This paper develops a New Bivariate-Gamma Distribution (NBGD) and explores its mathematical and statistical properties such as marginal probability distributions, moments, product moment, covariance and correlation. The study further investigates the various special cases of the NBGD. The new distribution is more robust with additional parameters and flexible for modelling real datasets.

Distribution of the Age of Gossip in Networks.

We study a general setting of gossip networks in which a source node forwards its measurements (in the form of status updates) about some observed physical process to a set of monitoring nodes according to independent Poisson processes. Furthermore, each monitoring node sends status updates about its information status (about the process observed by the source) to the other monitoring nodes according to independent Poisson processes. We quantify the freshness of the information available at each monitoring node in terms of Age of Information (AoI). While this setting has been analyzed in a handful of prior works, the focus has been on characterizing the average (i.e., marginal first moment) of each age process. In contrast, we aim to develop methods that allow the characterization of higher-order marginal or joint moments of the age processes in this setting. In particular, we first use the stochastic hybrid system (SHS) framework to develop methods that allow the characterization of the stationary marginal and joint moment generating functions (MGFs) of age processes in the network. These methods are then applied to derive the stationary marginal and joint MGFs in three different topologies of gossip networks, with which we derive closed-form expressions for marginal or joint high-order statistics of age processes, such as the variance of each age process and the correlation coefficients between all possible pairwise combinations of age processes. Our analytical results demonstrate the importance of incorporating the higher-order moments of age processes in the implementation and optimization of age-aware gossip networks rather than just relying on their average values.

A New Bivariate Odd Generalized Exponential Gompertz Distribution

The objective of this study was to present a novel bivariate distribution, which we denoted as the bivariate odd generalized exponential gompertz(BOGE-G) distribution. Other well-known models included in this one include the gompertz, generalized exponential, odd generalized exponential, and odd generalized exponential gompertz distribution. The model introduced here is of Marshall-Olkin type [16]. The marginals of the new bivariate distribution have odd generalized exponential gompertz distribution which proposed by[7]. Closed forms exist for both the joint probability density function and the joint cumulative distribution function. The bivariate moment generating function, marginal moment generating function, conditional distribution, joint reliability function, marginal hazard rate function, joint mean waiting time, and joint reversed hazard rate function are some of the properties of this distribution that have been discussed. The maximum likelihood approach is used to estimate the model parameters. To demonstrate empirically the significance and adaptability of the new model in fitting and evaluating real lifespan data, two sets of real data are studied using the new bivariate distribution. Using the software Mathcad, a simulation research was conducted to evaluate the bias and mean square error (MSE) characteristics of MLE. We found that the bias and MSE decrease as the sample size increases.

Nonparametric estimation of conditional marginal excess moments

Several risk measures have been proposed in the literature, among them the marginal mean excess, defined as MMEp=E[{Y(1)−Q1(1−p)}+|Y(2)>Q2(1−p)], provided E|Y(1)|<∞, where (Y(1),Y(2)) denotes a pair of risk factors, y+≔max(0,y), Qj the quantile function of Y(j),j∈{1,2}, and p∈(0,1). In this paper we consider a generalization of this measure, where the random variables of main interest (Y(1),Y(2)) are observed together with a random covariate X∈Rd, and where the Y(1) excess is also power transformed. This leads to the concept of conditional marginal excess moment for which an estimator is proposed allowing extrapolation outside the data range. The main asymptotic properties of this estimator have been established, using empirical processes arguments combined with the multivariate extreme value theory. The finite sample behavior of the estimator is evaluated by a simulation experiment. We apply also our method on a vehicle insurance customer dataset.

Wind-driven underwater ambient noise for seabed remote sensing: A review of methods and results

Estimation of seabed geoacoustic parameters in shallow water by acoustic remote sensing remains a challenging task due to constraints on hardware, data collection and analysis, and cost of maritime surveys. The work of Lisa Zurk and Martin Siderius, among many others, inspired investigation of signal processing techniques to process wind-driven underwater noise for the estimation of seabed geoacoustics. This paper presents a summary of the work on Bayesian geoacoustic inversion of ambient noise carried out at University of Victoria and Portland State University, using fixed and drifting vertical line arrays (VLA). The Bayesian inversion framework is based on Markov-chain Monte Carlo sampling. It estimates a joint posterior probability density function, from which marginal density functions, moments and covariances between geoacoustic parameters of interest can be quantified. The approach was applied to simulated and experimental ambient noise data for the estimation of seabed layer thicknesses, sound speeds, densities, and sediment attenuations. The resolution of the method was explored as a function of experiment factors such as array design and wind speed, and a method for extending the aperture of a VLA by extrapolating the noise coherence was proposed.

Projecting Flood Frequency Curves Under Near‐Term Climate Change

AbstractFlood‐frequency curves, critical for water infrastructure design, are typically developed based on a stationary climate assumption. However, climate changes are expected to violate this assumption. Here, we propose a new, climate‐informed methodology for estimating flood‐frequency curves under non‐stationary future climate conditions. The methodology develops an asynchronous, semiparametric local‐likelihood regression (ASLLR) model that relates moments of annual maximum flood to climate variables using the generalized linear model. We estimate the first two marginal moments (MM) – the mean and variance – of the underlying log‐Pearson Type‐3 distribution from the ASLLR with the monthly rainfall and temperature as predictors. The proposed methodology, ASLLR‐MM, is applied to 40 U.S. Geological Survey streamgages covering 18 water resources regions across the conterminous United States. A correction based on the aridity index was applied on the estimated variance, after which the ASLLR‐MM approach was evaluated with both historical (1951–2005) and projected (2006–2035, under RCP4.5 and RCP8.5) monthly precipitation and temperature from eight Global Circulation Models (GCMs) consisting of 39 ensemble members. The estimated flood‐frequency quantiles resulting from the ASLLR‐MM and GCM members compare well with the flood‐frequency quantiles estimated using the historical period of observed climate and flood information for humid basins, whereas the uncertainty in model estimates is higher in arid basins. Considering additional atmospheric and land‐surface conditions and a multi‐level model structure that includes other basins in a region could further improve the model performance in arid basins.

Stochastic Analysis of the Marginal and Dependence Structure of Streamflows: From Fine-Scale Records to Multi-Centennial Paleoclimatic Reconstructions

The identification of the second-order dependence structure of streamflow has been one of the oldest challenges in hydrological sciences, dating back to the pioneering work of H.E Hurst on the Nile River. Since then, several large-scale studies have investigated the temporal structure of streamflow spanning from the hourly to the climatic scale, covering multiple orders of magni-tude. In this study, we expanded this range to almost eight orders of magnitude by analysing small-scale streamflow time series (in the order of minutes) from ground stations and large-scale streamflow time series (in the order of hundreds of years) acquired from paleocli-matic reconstructions. We aimed to determine the fractal behaviour and the long-range de-pendence behaviour of the streamflow. Additionally, we assessed the behaviour of the first four marginal moments of each time series to test whether they follow similar behaviours as sug-gested in other studies in the literature. The results provide evidence in identifying a common stochastic structure for the streamflow process, based on the Pareto–Burr–Feller marginal dis-tribution and a generalized Hurst–Kolmogorov (HK) dependence structure.

A bimatrix variate gamma distribution

In this article, a bimatrix gamma distributions is introduced. Various mathematical properties of the proposed distribution like marginal distributions, expected values, entropies, and moment generating function are derived. Also, distributions of sum, quotient, and product are investigated. Parameter estimation by the method of maximum likelihood is considered. A simulation study is performed to check the finite sample performance of the maximum likelihood estimators.

Simulating multivariate stationary non-Gaussian process based on wavenumber–frequency spectrum and unified Hermite polynomial model

It is of importance to model the input excitations considering the spatial variation and non-Gaussianity for the reliability evaluation of long span structures. The existing multivariate stationary non-Gaussian simulation methods are mainly based on the target marginal distribution and cross-spectrum matrix. Due to the Cholesky decomposition of cross-spectrum matrix, the computational cost of these methods increases rapidly with the increase of spatial points. Therefore, this paper develops a novel simulation method based on the wavenumber–frequency spectrum and unified Hermite polynomial model (UHPM). Firstly, UHPM is extended to the homogenous and stationary non-Gaussian field. Then, a complete transformation model from homogenous and stationary non-Gaussian auto-correlation function (ACF) into underlying Gaussian ACF with its applicable range is proposed. Furthermore, two types of incompatibility between the first four marginal moments and wavenumber–frequency spectrum are discussed, and the corresponding remedies are provided. To facilitate the calculation, the 2D-Fast Fourier Transform (FFT) technique is embedded in the Wiener–Khintchine transformation and spectral representation method (SRM). Finally, a unified simulation framework for multivariate stationary non-Gaussian process based on its relationship to the homogenous and stationary non-Gaussian field is presented. Two numerical examples, involving the simulations of non-Gaussian wind velocities and ground motion accelerations, are investigated to verify the capabilities of the proposed method.

Constrained overdamped Langevin dynamics for symmetric multimarginal optimal transportation

The Strictly Correlated Electrons (SCE) limit of the Levy–Lieb functional in Density Functional Theory (DFT) gives rise to a symmetric multi-marginal optimal transport problem with Coulomb cost, where the number of marginal laws is equal to the number of electrons in the system, which can be very large in relevant applications. In this work, we design a numerical method, built upon constrained overdamped Langevin processes to solve Moment Constrained Optimal Transport (MCOT) relaxations (introduced in [A. Alfonsi, R. Coyaud, V. Ehrlacher and D. Lombardi, Approximation of optimal transport problems with marginal moments constraints, Math. Comp. 90 (2021) 689–737; C. Villani, Optimal Transport: Old and New (Springer Science &amp; Business Media, 2008)]) of symmetric multi-marginal optimal transport problems with Coulomb cost. Some minimizers of such relaxations can be written as discrete measures charging a low number of points belonging to a space whose dimension, in the symmetrical case, scales linearly with the number of marginal laws. We leverage the sparsity of those minimizers in the design of the numerical method and prove that there is no strict local minimizer to the resulting problem. We illustrate the performance of the proposed method by numerical examples which solves MCOT relaxations of 3D systems with up to 100 electrons.

Sampling discretization and related problems

This survey addresses sampling discretization and its connections with other areas of mathematics. The survey concentrates on sampling discretization of norms of elements of finite-dimensional subspaces. We present here known results on sampling discretization of both integral norms and the uniform norm beginning with classical results and ending with very recent achievements. We also show how sampling discretization connects to spectral properties and operator norms of submatrices, embedding of finite-dimensional subspaces, moments of marginals of high-dimensional distributions, and learning theory. Along with the corresponding results, important techniques for proving those results are discussed as well.