Kendall Distribution Research Articles

On convergence and mass distributions of multivariate Archimedean copulas and their interplay with the Williamson transform

Motivated by a recently established result saying that within the class of bivariate Archimedean copulas standard pointwise convergence implies weak convergence of almost all conditional distributions this contribution studies the class Card of all d-dimensional Archimedean copulas with d≥3 and proves the afore-mentioned implication with respect to conditioning on the first d−1 coordinates. Several properties equivalent to pointwise convergence in Card are established and - as by-product of working with conditional distributions (Markov kernels) - alternative simple proofs for the well-known formulas for the level set masses μC(Lt) and the Kendall distribution function FKd as well as a novel geometrical interpretation of the latter are provided. Viewing normalized generators ψ of d-dimensional Archimedean copulas from the perspective of their so-called Williamson measures γ on (0,∞) is then shown to allow not only to derive surprisingly simple expressions for μC(Lt) and FKd in terms of γ and to characterize pointwise convergence in Card by weak convergence of the Williamson measures but also to prove that regularity/singularity properties of γ directly carry over to the corresponding copula Cγ∈Card. These results are finally used to prove the fact that the family of all absolutely continuous and the family of all singular d-dimensional copulas is dense in Card and to underline that despite of their simple algebraic structure Archimedean copulas may exhibit surprisingly singular behavior in the sense of irregularity of their conditional distribution functions.

A weighted independence test based on smooth estimation of Kendall distribution

In this study, we introduce a nonparametric test of independence which is based on a weighted Cramér-von Mises distance of the Bernstein estimate of the Kendall distribution function. We examine the power and the size of the new weighted test and, we compare it with the non-weighted nonparametric tests of independence through a Monte Carlo simulation study. The simulation results indicate that both of the choices of the weights and the polynomial degree has a significant effect on the test power. Finally, the procedure is applied to a dataset on chemical elements in water samples for the purpose of illustration.

Copula-based modeling of hydraulic structures using a nonlinear reservoir model

Abstract Multivariate flood frequency analysis has been widely used in the design and risk assessment of hydraulic structures. However, analytical solutions are often obtained based on an idealized linear reservoir model in which a linear routing process is assumed, and consequently, the flood risk is likely to be over- or underestimated. The present study proposes a nonlinear reservoir model in which the relationships of reservoir water level with reservoir volume and discharge are assumed to be nonlinear in order to more accurately describe the routing process as it takes into consideration the interactions between hydrological loading and different discharge structures. The structure return period is calculated based on the copula function and compared with that based on the linear reservoir model and the bivariate return period based on the Kendall distribution function. The results show that the structure return period based on the linear model leads to an underestimation of the flood risk under the conditions of high reservoir water level. For the same reservoir, linear and nonlinear reservoir models give quite different reservoir volume-water level and discharge-water level curves; therefore, they differ substantially in the sensitivity to flood events with different combinations of flood peak and volume. We also analyze the effects of the parameters involved in the reservoir volume-water level and discharge-water level relationships on the maximum water level at different return periods in order to better understand the applicability and effectiveness of the proposed method for different hydraulic projects.

Drought propagation and construction of a comprehensive drought index based on the Soil and Water Assessment Tool (SWAT) and empirical Kendall distribution function (<i>K</i><sub>C′</sub>): a case study for the Jinta River basin in northwestern China

Abstract. Monitoring drought and mastering the laws of drought propagation are the basis for regional drought prevention and resistance. Multivariate drought indicators considering meteorological, agricultural and hydrological information may fully describe drought conditions. However, series of hydrological variables in cold and arid regions that are too short or missing make it difficult to monitor drought. This paper proposed a method combining Soil and Water Assessment Tool (SWAT) and empirical Kendall distribution function (KC′) for drought monitoring. The SWAT model, based on the principle of runoff formation, was used to simulate the hydrological variables of the drought evolution process. Three univariate drought indexes, namely meteorological drought (standardized precipitation evapotranspiration index; SPEI), agricultural drought (standardized soil moisture index; SSI) and hydrological drought (standardized streamflow drought index; SDI), were constructed using a parametric or non-parametric method to analyze the propagation time from meteorological drought to agricultural drought and hydrological drought. The KC′ was used to build a multivariable comprehensive meteorology–agriculture–hydrology drought index (MAHDI) that integrated meteorological, agricultural and hydrological drought to analyze the characteristics of a comprehensive drought evolution. The Jinta River in the inland basin of northwestern China was used as the study area. The results showed that agricultural and hydrological drought had a seasonal lag time from meteorological drought. The degree of drought in this basin was high in the northern and low in the southern regions. MAHDI proved to be acceptable in that it was consistent with historical drought records, could catch drought conditions characterized by univariate drought indexes, and capture the occurrence and end of droughts. Nevertheless, its ability to characterize mild and moderate droughts was stronger than severe droughts. In addition, the comprehensive drought conditions showed insignificant aggravating trends in spring and summer and showed insignificant alleviating trends in autumn and winter and at annual scales. The results provided theoretical support for the drought monitoring in the Jinta River basin. This method provided the possibility for drought monitoring in other watersheds lacking measured data.

A machine learning approach to galaxy properties: joint redshift–stellar mass probability distributions with Random Forest

ABSTRACTWe demonstrate that highly accurate joint redshift–stellar mass probability distribution functions (PDFs) can be obtained using the Random Forest (RF) machine learning (ML) algorithm, even with few photometric bands available. As an example, we use the Dark Energy Survey (DES), combined with the COSMOS2015 catalogue for redshifts and stellar masses. We build two ML models: one containing deep photometry in the griz bands, and the second reflecting the photometric scatter present in the main DES survey, with carefully constructed representative training data in each case. We validate our joint PDFs for 10 699 test galaxies by utilizing the copula probability integral transform and the Kendall distribution function, and their univariate counterparts to validate the marginals. Benchmarked against a basic set-up of the template-fitting code bagpipes, our ML-based method outperforms template fitting on all of our predefined performance metrics. In addition to accuracy, the RF is extremely fast, able to compute joint PDFs for a million galaxies in just under 6 min with consumer computer hardware. Such speed enables PDFs to be derived in real time within analysis codes, solving potential storage issues. As part of this work we have developed galpro1, a highly intuitive and efficient python package to rapidly generate multivariate PDFs on-the-fly. galpro is documented and available for researchers to use in their cosmology and galaxy evolution studies.

Multivariate return period‐based ground motion selection for improved hazard consistency over a vector of intensity measures

SummaryGround motion selection is a crucial step in probabilistic seismic performance assessment of structural systems. Particularly, identifying ground motion records compatible with a specific hazard level has been the major focus of past studies. We propose a multivariate return period (MRP)‐based record selection methodology to further improve the multivariate hazard consistency over a vector of intensity measures (IMs). Unlike the traditional univariate return period anchored on a scalar IM, MRP generalizes the return period concept by accommodating the joint rate of exceedance of a vector of IMs, thereby providing more holistic seismic hazard characterization. By leveraging MRP in linking the seismic hazard to a vector IMs, the proposed MRP‐based ground motion selection methodology for the first time offers a mathematically rigorous yet practical solution for multivariate hazard consistency in ground motion selection. The merit of the MRP‐based ground motion selection is demonstrated by comparing the resulting target spectra and seismic demand estimates for several case‐study structures with other state‐of‐the‐art ground motion selection alternatives. From the results, the MRP‐based ground motion selection employing Kendall's distribution function turns out to be a promising alternative, offering favorable new features including (a) moderate target spectra intensity and moderately low target spectra standard deviation; (b) superior convergence with the increase of the conditioning IM dimension and ability to approximate higher‐dimensional multivariate hazard consistency with lower dimensional conditioning IMs; and (c) capability to realistically capture the multimodal spectral shape owing to the incorporation of multivariate Gaussian mixture distribution in generating target spectra.

BIVARIATE MARSHALL–OLKIN EXPONENTIAL SHOCK MODEL

AbstractThe well-known Marshall–Olkin model is known for its extension of exponential distribution preserving lack of memory property. Based on shock models, a new generalization of the bivariate Marshall–Olkin exponential distribution is given. The proposed model allows wider range tail dependence which is appealing in modeling risky events. Moreover, a stochastic comparison according to this shock model and also some properties, such as association measures, tail dependence and Kendall distribution, are presented. The new shock model is analytically quite tractable, and it can be used quite effectively, to analyze discrete–continuous data. This has been shown on real data. Finally, we propose the multivariate extension of the Marshall–Olkin model that has some intersection with the well-known multivariate Archimax copulas.

A goodness-of-fit test based on Bézier curve estimation of Kendall distribution

ABSTRACTIn this study, we propose an estimation method for the Archimedean family of the copula in a nonparametric setting. A Bézier curve approach based on Bernstein polynomials is used to estimate the Kendall distribution function. Also, a new goodness-of-fit test based on Cramér–von Mises statistic is proposed using the Bézier curve estimator. A Monte Carlo study is also conducted to measure the performance of the proposed estimator and goodness-of-fit test. Two real data examples are also given. The simulation study and real data applications show that the Bézier curve estimator leads to satisfactory estimates of underlying copula and also has better results compared with the estimators based on empirical and Bernstein methods.

Relations between ageing and dependence for exchangeable lifetimes with an extension for the IFRA/DFRA property

AbstractWe first review an approach that had been developed in the past years to introduce concepts of “bivariate ageing” for exchangeable lifetimes and to analyze mutual relations among stochastic dependence, univariate ageing, and bivariate ageing.A specific feature of such an approach dwells on the concept of semi-copula and in the extension, from copulas to semi-copulas, of properties of stochastic dependence. In this perspective, we aim to discuss some intricate aspects of conceptual character and to provide the readers with pertinent remarks from a Bayesian Statistics standpoint. In particular we will discuss the role of extensions of dependence properties. “Archimedean” models have an important role in the present framework.In the second part of the paper, the definitions of Kendall distribution and of Kendall equivalence classes will be extended to semi-copulas and related properties will be analyzed. On such a basis, we will consider the notion of “Pseudo-Archimedean” models and extend to them the analysis of the relations between the ageing notions of IFRA/DFRA-type and the dependence concepts of PKD/NKD.

Lorenz-generated bivariate Archimedean copulas

Abstract A novel generating mechanism for non-strict bivariate Archimedean copulas via the Lorenz curve of a non-negative random variable is proposed. Lorenz curves have been extensively studied in economics and statistics to characterize wealth inequality and tail risk. In this paper, these curves are seen as integral transforms generating increasing convex functions in the unit square. Many of the properties of these “Lorenz copulas”, from tail dependence and stochastic ordering, to their Kendall distribution function and the size of the singular part, depend on simple features of the random variable associated to the generating Lorenz curve. For instance, by selecting random variables with a lower bound at zero it is possible to create copulas with asymptotic upper tail dependence. An “alchemy” of Lorenz curves that can be used as general framework to build multiparametric families of copulas is also discussed.

A framework for measuring association of random vectors via collapsed random variables

A framework for quantifying dependence between random vectors is introduced. Using the notion of a collapsing function, random vectors are summarized by single random variables, referred to as collapsed random variables. Measures of association computed from the collapsed random variables are then used to measure the dependence between random vectors. To this end, suitable collapsing functions are presented. Furthermore, the notion of a collapsed distribution function and collapsed copula are introduced and investigated for certain collapsing functions. This investigation yields a multivariate extension of the Kendall distribution and its corresponding Kendall copula for which some properties and examples are provided. In addition, non-parametric estimators for the collapsed measures of association are provided along with their corresponding asymptotic properties. Finally, data applications to bioinformatics and finance are presented along with a general graphical assessment of independence between groups of random variables.

Lorenz-Generated Bivariate Archimedean Copulas

An alternative generating mechanism for non-strict bivariate Archimedean copulas via the curve of a positive random variable is proposed. curves have been extensively studied in economics and statistics to characterize wealth inequality and tail risk. In this paper, these curves are seen as integral transforms generating increasing convex functions in the unit square. Many of the properties of these Lorenz copulas, from tail dependence and stochastic ordering, to their Kendall distribution function and the size of the singular part, depend on simple features of the random variable associated to the generating curve. For instance, by selecting random variables with lower bound at zero it is possible to create copulas with asymptotic upper tail dependence. An alchemy of curves that can be used as general framework to build multiparametric families of copulas is also discussed.

Multi-Panel Kendall plot in light of an ROC curve analysis applied to measuring dependence

ABSTRACTThe Kendall plot (-plot) is a plot measuring dependence between the components of a bivariate random variable. The -plot graphs the Kendall distribution function against the distribution function of VU, where V and U are independent uniform random variables. We associate -plots with the receiver operating characteristic () curve, a well-accepted graphical tool in biostatistics for evaluating the ability of a biomarker to discriminate between two populations. The most commonly used global index of diagnostic accuracy of biomarkers is the area under the curve (). In parallel with the , we propose a novel strategy to measure association between random variables from a continuous bivariate distribution. First, we discuss why the area under the conventional Kendall curve () cannot be used as an index of dependence. We then suggest a simple and meaningful extension of the definition of the -plots, and define an index of dependence that is based on . This measure characterizes a wide range of two-variable relationships, thereby completely detecting the underlying dependence structure. Properties of the proposed index satisfy the mathematical definition of a measure. Finally, simulations and real data examples illustrate the applicability of the proposed method.

Testing independence for Archimedean copula based on Bernstein estimate of Kendall distribution function

ABSTRACTIn this study, we estimate the Kendall distribution function for Archimedean copula family using Bernstein polynomial approximation and we investigate its performance by Monte Carlo simulation. Then, we introduce a nonparametric test of independence which is based on Cramér–von-Mises distance of the new estimate of Kendall distribution function. Also, we examine the power and the size of the test and we compare it with the classical nonparametric test that is based on the empirical Kendall distribution function.

The analysis of left truncated bivariate data using frailty models

AbstractIn this paper, we study the properties of frailty models for bivariate data under fixed left truncation. It turns out that the distribution of completely observable pairs is also a member of the Archimedean copula family. We propose a new estimation strategy to analyze this type of data using the corresponding Kendall distribution. A general goodness‐of‐fit test procedure is then developed for selecting the best model. Our strategies are generalization of the methodologies proposed for left truncated data. We demonstrate our new approaches using simulations and an illustrative example and end our paper with some discussion.

Archimedean Copula Estimation Parameter with Kendall Distribution Function

In the literature, up to now, it is common that for Gumbel, Clayton and Frank calculated Kendall Distribution function and to the extent those applications have been made. In this paper, we made Kendall Distribution function calculation for Ali Mikhail Haq and Joe and in relation that simulation study. We generated dependent gamma distribution. For dependency between these variables we used Archimedean copula. In connection with this, we define basic properties of copulas and their nonparametric method. In this study, to explain the relationship between the variables, five Archimedean copula families were used; Gumbel, Clayton, Frank Joe and Ali Mikhail Haq. We obtained nonparametric estimation of these copula families parameters and the suitable Archimedean copula family for this data set.

Archimedean Copula Parameter Estimation with Kendall Distribution Function

In the literature, up to now, it is common that for Gumbel, Clayton and Frank calculated Kendall Distribution function K u ( ) and to the extent those applications have been made. Kendall distribution functions show stochastic orderings of random vectors. The aim of Kendall distribution function is selected suitable copula function for using data set. For dependence structures of the data set, we calculated Kendall Tau and Spearman Rho values which are nonparametric. Based on this method, parameters of copula are obtained. In this paper, we are made Kendall Distribution function which obtained with the help of generator function of Archimedean copula calculation for Ali Mikhail Haq and Joe and in relation with that simulation study. We used data set which generated dependent generalized pareto distribution (Gp(3,3,3)) for this study. For dependency among these variables, we used Archimedean copula. In connection with this, we defne basic properties of copulas and nonparametric methods Kendall Tau, Spearman Rho are given. In this study, to explain the relationship among the variables, fve Archimedean copula are selected; Gumbel, Clayton, Frank Joe and Ali Mikhail Haq. Afterwards, we are obtained nonparametric estimation of parameters of these copulas with the help of Kendall Tau. With Kendall distribution function values, we found the suitable Archimedean copula family for this data set.

Bivariate return periods and their importance for flood peak and volume estimation

Estimates of flood event magnitudes with a certain return period are required for the design of hydraulic structures. While the return period is clearly defined in a univariate context, its definition is more challenging when the problem at hand requires considering the dependence between two or more variables in a multivariate framework. Several ways of defining a multivariate return period have been proposed in the literature, which all rely on different probability concepts. Definitions use the conditional probability, the joint probability, or can be based on the Kendall's distribution or survival function. In this study, we give a comprehensive overview on the tools that are available to define a return period in a multivariate context. We especially address engineers, practitioners, and people who are new to the topic and provide them with an accessible introduction to the topic. We outline the theoretical background that is needed when one is in a multivariate setting and present the reader with different definitions for a bivariate return period. Here, we focus on flood events and the different probability concepts are explained with a pedagogical, illustrative example of a flood event characterized by the two variables peak discharge and flood volume. The choice of the return period has an important effect on the magnitude of the design variable quantiles, which is illustrated with a case study in Switzerland. However, this choice is not arbitrary and depends on the problem at hand. WIREs Water 2016, 3:819–833. doi: 10.1002/wat2.1173This article is categorized under: Engineering Water > Methods Engineering Water > Planning Water Science of Water > Water Extremes

Singularity aspects of Archimedean copulas

Calculating Markov kernels of two-dimensional Archimedean copulas allows for very simple and elegant alternative derivations of various important formulas including Kendall's distribution function and the measures of the level curves. More importantly, using Markov kernels we prove the existence of singular Archimedean copulas Aφ with full support of the following two types: (i) All conditional distribution functions y↦FxAφ(y) are discrete and strictly increasing; (ii) all conditional distribution functions y↦FxAφ(y) are continuous, strictly increasing and have derivative zero almost everywhere. The results show that despite of their simple analytic form Archimedean copulas can exhibit surprisingly singular behavior.

Hierarchical Kendall copulas: Properties and inference

AbstractWhile there is substantial need for dependence models in higher dimensions, most existing models quickly become rather restrictive and barely balance parsimony and flexibility. Hierarchical constructions may improve on that by grouping variables in different levels. In this paper, the new class of hierarchical Kendall copulas is proposed and discussed. Hierarchical Kendall copulas are built up by flexible copulas specified for groups of variables, where aggregation is facilitated by the Kendall distribution function, the multivariate analog to the probability integral transform for univariate random variables. After deriving properties of the general model formulation, particular focus is given to inference techniques of hierarchical Kendall copulas with Archimedean components, for which closed‐form analytical expressions can be derived. A substantive application to German stock returns finally shows that hierarchical Kendall copulas perform very well for real data, out‐of‐ as well as in‐sample. The Canadian Journal of Statistics 42: 78–108; 2014 © 2014 Statistical Society of Canada