Continuous Marginal Distributions Research Articles

VEGETATED HYDRODYNAMIC SYSTEM: PARAMETERIZATION AND STOCHASTIC DEPENDENCE MODELLING

Vegetation as a nature-based solution for increasing flood risk has convincingly shown potential for flood hazard (wave load) reduction but lacks generalized results. In this study we have introduced stochastic dependence modeling using non-parametric Bayesian networks (NPBN) for vegetated coastal systems where the system was parametrized using continuous marginal distributions, and likely (conditional) correlations among variables. The model represented a consistent joint probability distribution and hence can be used to generate physically realistic conditions in data-scare environments. It adds value to numerical modeling by reducing the number of simulations required to get meaningful generalized results. Main findings, that were derived by using a NPBN, help to pave way for implementation of nature-based solutions for a range of realistic conditions that can be found across global coastal foreshores.Recorded Presentation from the vICCE (YouTube Link): https://youtu.be/T6TP0DH0qMw

A copula transformation in multivariate mixed discrete-continuous models

Copulas allow a flexible and simultaneous modeling of complicated dependence structures together with various marginal distributions. Especially if the density function can be represented as the product of the marginal density functions and the copula density function, this leads to both an intuitive interpretation of the conditional distribution and convenient estimation procedures. However, this is no longer the case for copula models with mixed discrete and continuous marginal distributions, because the corresponding density function cannot be decomposed so nicely. In this paper, we introduce a copula transformation method that allows to represent the density function of a distribution with mixed discrete and continuous marginals as the product of the marginal probability mass/density functions and the copula density function. With the proposed method, conditional distributions can be described analytically and the computational complexity in the estimation procedure can be reduced depending on the type of copula used.

Frequency analyses of peak discharge and suspended sediment concentration in the United States

Compound extreme events with concurrent peak discharges and suspended sediment concentrations have substantial impacts such as floods, infrastructure failure, pollution, increased water treatment costs, and threats to aquatic life. It is thus essential to understand the spatiotemporal dynamics of water and sediment transport during such extreme flood events. Copulas are used to conduct multivariate flood frequency analyses and provide relationships between design variables and recurrence intervals that are useful in hydrology and water resource management. We used hydrological data from 20 gauging stations on important rivers across the United States with daily records spanning at least the last 30 years. We extracted annual maximum discharges (Qmax) and corresponding suspended sediment concentrations (SSCmax) at these 20 stations. We analyzed the temporal trends of Qmax and SSCmax by using the Mann–Kendall non-parametric statistical test. Then, we investigated the interdependency of Qmax and SSCmax through various correlation coefficients and used the Multivariate Copula Analysis Toolbox (MvCAT) for flood frequency analyses. We found Qmax decreased significantly for one river, whereas SSCmax decreased significantly for 10 of the 20 rivers over the analysis period. Five of rivers selected as statistically suitable for copula analysis were of the smallest size and with the fewest dams. Seven of rivers showed significantly decreasing SSCmax trends were of the largest size and with the most dams. The remaining eight rivers were of moderate size. In larger watersheds, the number of dams more strongly affected sediment transport dynamics. Soil condition and forest cover exerted stronger influences on SSCmax in smaller than in medium-sized watersheds, and soil condition was better correlated with the spatial SSCmax distribution than forest cover. Using MvCAT, we computed the optimal continuous marginal distribution functions for the five suitable rivers, and the optimal copula was different in each case. Our results indicate reduced sediment transport during extreme flood events. Copula functions are very useful in such flood frequency analyses, and measures should be applied according to local conditions to achieve optimal water resource management and planning.

Sums of standard uniform random variables

AbstractIn this paper, we analyse the set of all possible aggregate distributions of the sum of standard uniform random variables, a simply stated yet challenging problem in the literature of distributions with given margins. Our main results are obtained for two distinct cases. In the case of dimension two, we obtain four partial characterization results. For dimension greater than or equal to three, we obtain a full characterization of the set of aggregate distributions, which is the first complete characterization result of this type in the literature for any choice of continuous marginal distributions.

Testing the symmetry of a dependence structure with a characteristic function

AbstractThis paper proposes competing procedures to the tests of symmetry for bivariate copulas of Genest, Nešlehová and Quessy (2012). To this end, the null hypothesis of symmetry is expressed in terms of the copula characteristic function that uniquely determines the copula of a given bivariate population with continuous marginal distributions. Then, test statistics based on L2 weighted distances computed from an empirical version of the copula characteristic function are proposed. Their asymptotic behavior is derived under the null hypothesis as well as under general alternatives. In particular, it is established that these rank statistics behave asymptotically as first-order degenerate V-statistics under the null hypothesis and this large-sample representation is exploited in order to provide suitably adapted multiplier bootstrap versions for the computation of p-values. The simulations that are reported show that the new tests are more powerful than the competing methods based on the empirical copula introduced by Genest, Nešlehová and Quessy (2012).

Mixed binary-continuous copula regression models with application to adverse birth outcomes.

Bivariate copula regression allows for the flexible combination of two arbitrary, continuous marginal distributions with regression effects being placed on potentially all parameters of the resulting bivariate joint response distribution. Motivated by the risk factors for adverse birth outcomes, many of which are dichotomous, we consider mixed binary-continuous responses that extend the bivariate continuous framework to the situation where one response variable is discrete (more precisely, binary) whereas the other response remains continuous. Utilizing the latent continuous representation of binary regression models, we implement a penalized likelihood-based approach for the resulting class of copula regression models and employ it in the context of modeling gestational age and the presence/absence of low birth weight. The analysis demonstrates the advantage of the flexible specification of regression impacts including nonlinear effects of continuous covariates and spatial effects. Our results imply that racial and spatial inequalities in the risk factors for infant mortality are even greater than previously suggested.

Distributionally Robust Linear and Discrete Optimization with Marginals

In this paper, we study the class of linear and discrete optimization problems in which the objective coefficients are chosen randomly from a distribution, and the goal is to evaluate robust bounds on the expected optimal value as well as the marginal distribution of the optimal solution. The set of joint distributions is assumed to be specified up to only the marginal distributions. We generalize the primal-dual formulations for this problem from the set of joint distributions with absolutely continuous marginal distributions to arbitrary marginal distributions using techniques from optimal transport theory. While the robust bound is shown to be NP-hard to compute for linear optimization problems, we identify a sufficient condition for polynomial time solvability using extended formulations. This generalizes the known tractability results under marginal information from 0-1 polytopes to a class of integral polytopes and has implications on the solvability of distributionally robust optimization problems in areas such as scheduling which we discuss.

Sequential motif profile of natural visibility graphs.

The concept of sequential visibility graph motifs-subgraphs appearing with characteristic frequencies in the visibility graphs associated to time series-has been advanced recently along with a theoretical framework to compute analytically the motif profiles associated to horizontal visibility graphs (HVGs). Here we develop a theory to compute the profile of sequential visibility graph motifs in the context of natural visibility graphs (VGs). This theory gives exact results for deterministic aperiodic processes with a smooth invariant density or stochastic processes that fulfill the Markov property and have a continuous marginal distribution. The framework also allows for a linear time numerical estimation in the case of empirical time series. A comparison between the HVG and the VG case (including evaluation of their robustness for short series polluted with measurement noise) is also presented.

G/G/∞ queues with renewal alternating interruptions

AbstractWe study G/G/∞ queues with renewal alternating service interruptions, where the service station experiences `up' and `down' periods. The system operates normally in the up periods, and all servers stop functioning while customers continue entering the system during the down periods. The amount of service a customer has received when an interruption occurs will be conserved and the service will resume when the down period ends. We use a two-parameter process to describe the system dynamics:Xr(t,y) tracking the number of customers in the system at timetthat have residual service times strictly greater thany. The service times are assumed to satisfy either of the two conditions: they are independent and identically distributed with a distribution of a finite support, or are a stationary and weakly dependent sequence satisfying the ϕ-mixing condition and having a continuous marginal distribution function. We consider the system in a heavy-traffic asymptotic regime where the arrival rate gets large and service time distribution is fixed, and the interruption down times are asymptotically negligible while the up times are of the same order as the service times. We show the functional law of large numbers and functional central limit theorem (FCLT) for the processXr(t,y) in this regime, where the convergence is in the space 𝔻([0,∞), (𝔻,L1)) endowed with the SkorokhodM1topology. The limit processes in the FCLT possess a stochastic decomposition property.

Modified inference function for margins for the bivariate clayton copula-based SUN Tobit Model

ABSTRACTThis paper extends the analysis of the bivariate Seemingly Unrelated Regression (SUN) Tobit model by modeling its nonlinear dependence structure through the Clayton copula. The ability to capture/model the lower tail dependence of the SUN Tobit model where some data are censored (generally, left-censored at zero) is an useful feature of the Clayton copula. We propose a modified version of the (classical) Inference Function for Margins (IFS) method by Joe and XP [H. Joe and J.J. XP, The estimation method of inference functions for margins for multivariate models, Tech. Rep. 166, Department of Statistics, University of British Columbia, 1996], which we refer to as Modified Inference Function for Margins (MIFF) method, to obtain the (point) estimates of the marginal and Clayton copula parameters. More specifically, we employ the (frequenting) data augmentation technique at the second stage of the IFS method (the first stage of the MIFF method is equivalent to the first stage of the IFS method) to generate the censored observations and then estimate the Clayton copula parameter. This process (data augmentation and copula parameter estimation) is repeated until convergence. Such modification at the second stage of the usual estimation method is justified in order to obtain continuous marginal distributions, which ensures the uniqueness of the resulting Clayton copula, as stated by Solar's [A. Solar, Fonctions de répartition à n dimensions et leurs marges, Publ. de l'Institut de Statistique de l'Université de Paris 8 (1959), pp. 229–231] theorem; and also to provide an unbiased estimate of the association parameter (the IFS method provides a biased estimate of the Clayton copula parameter in the presence of censored observations in both margins). Since the usual asymptotic approach, that is the computation of the asymptotic covariance matrix of the parameter estimates, is troublesome in this case, we also propose the use of resampling procedures (bootstrap methods, such as standard normal and percentile, by Efron and Tibshirani [B. Efron and R.J. Tibshirani, An Introduction to the Bootstrap, Chapman & Hall, New York, 1993] to obtain confidence intervals for the model parameters.

Modelling Poisson marked point processes using bivariate mixture transition distributions

A class of bivariate continuous-discrete distributions is proposed to fit Poisson dynamic models in a single unified framework via bivariate mixture transition distributions (BMTDs). Potential advantages of this class over the current models include its ability to capture stretches, bursts and nonlinear patterns characterized by Internet network traffic, high-frequency financial data and many others. It models the inter-arrival times and the number of arrivals (marks) in a single unified model which benefits from the dependence structure of the data. The continuous marginal distributions of this class include as special cases the exponential, gamma, Weibull and Rayleigh distributions (for the inter-arrival times), whereas the discrete marginal distributions are geometric and negative binomial. The conditional distributions are Poisson and Erlang. Maximum-likelihood estimation is discussed and parameter estimates are obtained using an expectation–maximization algorithm, while the standard errors are estimated using the missing information principle. It is shown via real data examples that the proposed BMTD models appear to capture data features better than other competing models.

A functional central limit theorem for empirical processes under a strong mixing condition

This paper introduces a functional central limit theorem for empirical processes endowed with real values from a strictly stationary random field that satisfies an interlaced mixing condition. We proceed by using a common technique from Billingsley (Convergence of probability measures, Wiley, New York, 1999), by first obtaining the limit theorem for the case where the random variables of the strictly stationary ρ′-mixing random field are uniformly distributed on the interval [0, 1]. We then generalize the result to the case where the absolutely continuous marginal distribution function is not longer uniform. In this case we show that the empirical process endowed with values from the ρ′-mixing stationary random field, due to the strong mixing condition, doesn’t converge in distribution to a Brownian bridge, but to a continuous Gaussian process with mean zero and the covariance given by the limit of the covariance of the empirical process. The argument for the general case holds similarly by the application of a standard variant of a result of Billingsley (1999) for the space D(−∞, ∞).

Copula models for insurance claim numbers with excess zeros and time-dependence

This paper develops two copula models for fitting the insurance claim numbers with excess zeros and time-dependence. The joint distribution of the claims in two successive periods is modeled by a copula with discrete or continuous marginal distributions. The first model fits two successive claims by a bivariate copula with discrete marginal distributions. In the second model, a copula is used to model the random effects of the conjoint numbers of successive claims with continuous marginal distributions. Zero-inflated phenomenon is taken into account in the above copula models. The maximum likelihood is applied to estimate the parameters of the discrete copula model. A two-step procedure is proposed to estimate the parameters in the second model, with the first step to estimate the marginals, followed by the second step to estimate the unobserved random effect variables and the copula parameter. Simulations are performed to assess the proposed models and methodologies.

Accounting for Parameter Uncertainty in Large-Scale Stochastic Simulations with Correlated Inputs

This paper considers large-scale stochastic simulations with correlated inputs having normal-to-anything (NORTA) distributions with arbitrary continuous marginal distributions. Examples of correlated inputs include processing times of workpieces across several workcenters in manufacturing facilities and product demands and exchange rates in global supply chains. Our goal is to obtain mean performance measures and confidence intervals for simulations with such correlated inputs by accounting for the uncertainty around the NORTA distribution parameters estimated from finite historical input data. This type of uncertainty is known as the parameter uncertainty in the discrete-event stochastic simulation literature. We demonstrate how to capture parameter uncertainty with a Bayesian model that uses Sklar's marginal-copula representation and Cooke's copula-vine specification for sampling the parameters of the NORTA distribution. The development of such a Bayesian model well suited for handling many correlated inputs is the primary contribution of this paper. We incorporate the Bayesian model into the simulation replication algorithm for the joint representation of stochastic uncertainty and parameter uncertainty in the mean performance estimate and the confidence interval. We show that our model improves both the consistency of the mean line-item fill-rate estimates and the coverage of the confidence intervals in multiproduct inventory simulations with correlated demands.

Exponential convergence in probability for empirical means of Lévy processes

Let (Xt)t⩾0 be a Levy process taking values in ℝd with absolutely continuous marginal distributions. Given a real measurable function f on ℝd in Kato’s class, we show that the empirical mean \( \frac{1} {t}\int_0^t {f(X_s )ds} \)f(Xs)ds converges to a constant z in probability with an exponential rate if and only if f has a uniform mean z. This result improves a classical result of Kahane et al. and generalizes a similar result of L. Wu from the Brownian Motion to general Levy processes.

Tail asymptotics for the sum of two heavy-tailed dependent risks

Let X 1 , X 2 denote positive heavy-tailed random variables with continuous marginal distribution functions F 1 and F 2, respectively. The asymptotic behavior of the tail of X 1 +X 2 is studied in a general copula framework and some bounds and extremal properties are provided. For more specific assumptions on F 1 , F 2 and the underlying dependence structure of X 1 and X 2, we survey explicit asymptotic results available in the literature and add several new cases.

Distribution functions of copulas: a class of bivariate probability integral transforms

We discuss a two-dimensional analog of the probability integral transform for bivariate distribution functions H 1 and H 2, i.e., the distribution function of the random variable H 1( X, Y) given that the joint distribution function of the random variables X and Y is H 2. We study the case when H 1 and H 2 have the same continuous marginal distributions, showing that the distribution function of H 1( X, Y) depends only on the copulas C 1 and C 2 associated with H 1 and H 2. We examine various properties of these “distribution functions of copulas”, and illustrate applications including dependence orderings and measures of association.

The empirical process of a short-range dependent stationary sequence under Gaussian subordination

Consider the stationary sequenceX 1=G(Z 1),X 2=G(Z 2),..., whereG(·) is an arbitrary Borel function andZ 1,Z 2,... is a mean-zero stationary Gaussian sequence with covariance functionr(k)=E(Z 1 Z k+1) satisfyingr(0)=1 and ∑ =1 ∞ |r(k)| m < ∞, where, withI{·} denoting the indicator function andF(·) the continuous marginal distribution function of the sequence {X n }, the integerm is the Hermite rank of the family {I{G(·)≦ x} −F(x):x∈R}. LetF n (·) be the empirical distribution function ofX 1,...,X n . We prove that, asn→∞, the empirical processn 1/2{F n (·)-F(·)} converges in distribution to a Gaussian process in the spaceD[−∞,∞].

General definition and sample counterparts of monotonic dependence functions of bivariate distributions

The notation of the monotonic dependence function μ X,Y, introduced in [14] as a function-valued parameter of a class of bivariate distributions with continuous marginal distribution functions is generalized to the case when the assumption of continuity is dropped. Fundamental properties charecterizing μX,Y as a proper measure of monotonic dependence are proved. Moreover a sample counterpart introduced and shown to converege to μX,Y with probability 1.

Estimation of the Medians for Dependent Variables

Joint intervals of bounded confidence are suggested for the medians of a bivariate population with continuous marginal distributions. The two intervals are of the classic type based on sample order statistics.