A copula is a multivariate distribution with uniform marginal distributions. Copulas are mostly used to represent or to model the structure of dependence between random variables, separately from the marginal distributions.

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Copula fitting in R

I want to fit Archimedean copula to my data based on 'ML' or 'MSE' procedures. The common way is that these procedures choose the best based on the similarity of copula values to empirical ones. I ...
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Estimating correlation(covariance) matrix when fitting a copula using R copula package [migrated]

I have a question about the R package copula. When using fitCopula to fit a copula to data, ...
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How to AB test two copulas?

Say that I model two seperate multivariate copula distirbutions in an experimental design. What is a statistical test that I can use to determine whether they are different?
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Simulating from normal copula in R with varying correlation matrices [migrated]

I'm trying to simulate correlated normal copula realizations using the copula package in R. I have estimated the varying correlation matrices and stored them in an ...
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Robustness of Gaussian Copula

I am looking for studies regarding the robustness of a multivariate Gaussian copula. Specifically I am wondering whether estimates of the dependence parameter in a multivariate Gaussian Copula (Sigma) ...
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How many observations to estimate a parameter of an Archimedean copula?

Let's consider for example the bivariate Gumbel copula. $$C(u_1, u_2)=\exp \left[-\left(\left(-\ln\left(u_1\right)\right)^{\theta}+\left(-\ln\left(u_2\right)\right)^{\theta}\right)^{\frac{1}{\theta}}\...
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1answer
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uniform histograms in copula approach

I would like to model some time series. For this purpose I have the marginal distribution and want to use a gaussian copula to build the dependency. Following a tutorial the following should give me ...
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Forming distribution conditioned on many variables from single conditional marginals using copulas

I'm brainstorming about a data analysis project, part of which can be thought of as estimating a joint distribution from marginals, so I'd like to know whether I can use some copula techniques. ...
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41 views

Tail dependence of mixture copulas

I am currently using (multivariate) mixture copulas to model a financial data set. The mixture has two components as follows: $$C_{mixture}=wC_1+(1-w)C_2$$ where $C_1$ and $C_2$ are copulas. I have ...
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Kendall's tau for extreme copulas: extreme-t and Hüsler-Reiss

I am looking for formulas to calculate Kendall's tau ($\tau_\mathrm{C}$) from copula parameter (or vice versa). I am interested in extreme-t and Hüsler-Reiss copulas. It is not an issue if numerical ...
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71 views

Time varying copula vs. dynamic copula

I need to understand the difference between time varying copula (Patton, 2006) and dynamic copula. For the time varying copula, is it when the parameters of copula follow ARMA($p$,$q$)? Is that ...
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Can I always analytically derive a Copula Function?

Say I have two Independent Marginals (say two poisson variables and one Gaussian). Can I analytically derive the Copula Function that relates all these variables? I'd only like a brief response. I'm ...
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Patton's dynamic Archimedean copula

I'm reading this paper by A. J. Patton http://www.christoffersen.com/CHRISTOP/2007/Patton_IER_2006.pdf on dynamic copulas whose parameter is driven, indirectly via Kendall's tau, by some ARMA-type ...
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How to simulate from a t copula?

This is a question related to: How to simulate from a Gaussian copula? Suppose that I have two univariate marginal distributions, say $F$ and $G$, which I can simulate from. Now, construct their ...
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how to estimate the coefficients of two margins by copula?

I am trying to fit a copula for two categorical variables(each variable in 4 categories) I=4 and J=4 If I plug margins and copula function into log likelihood, and apply optim, I can estimate the ...
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Multivariate copula estimation

Multivariate copulas are oftentimes used for financial modeling. A copula can be constructing by tying together a series of marginal distributions, with a dependency structure. In case of a t-copula, ...
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zipf and correlated lognormal

I have been struggling with this for a while. I want to generate two random variables $X$ and $Y$ with a particular correlation $\rho$ where $X$ is the file popularity (zipf distribution) and $Y$ is ...
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1answer
109 views

How to construct a bivariate distribution from marginal distributions with a predefined correlation

I would like to generate zipf and lognormal random variables with a particular correlation. Then, I would like to find their bivariate distribution. What approach should I follow?.
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Choose a Copula for multivariate data

I want to construct a Copula to obtain a PDF that includes the dependence in the variables. Each random variable is uniformly distributed and the covariance/correlation matrix is known. I don't have ...
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1answer
56 views

probabilities with copula

i am trying to calculate the probability of tow random continuous variable being in an interval with the help of copula i am starting with i want to calculate it with R but i dont know how to do ...
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Copula density bounds

I was wondering if someone can help me with a problem I encountered in my work. I need a bivariate copula density that meets two constraints at the bounds, and I have difficulties in finding one that ...
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2answers
48 views

How do we separate marginals from dependence using copulas, and why do we assume uniform marginals?

I read that one advantage of using copula function is that, we can separate the marginals from dependence between variables. I tried to understand how we can do that but I could not find the answer. ...
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1answer
80 views

Partial Derivative of Joint Distribution Function interpretation

Suppose we have \begin{equation} F(x,y) = \int_{-\infty}^x \int_{-\infty}^y f(a,b) \ db \ da \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [1] \end{equation} From this, we can say the following: \begin{align} \...
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Empirical multivariate probability integral transform

Is there a 'simple' way to obtain a non-parametric empirical multivariate probability integral transform? Univariate case The probability integral transform relates to the transform of any random ...
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55 views

Joint Probability Density Function

I'm trying to find the Joint Probability Density Function of three variables, they are random continuous variables. I was thinking about using a copula function since I saw them recommended here ...
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Simulate a Gaussian Copula with t margins

The task is the following: Given is $Z_1,...Z_{50}$ different hypothetical assets. Each $Z_k \sim t_3$ with standard deviation $\sigma=0.01$ and $\tau(Z_i,Z_k)=0.4$ for $j\neq k$. I want to ...
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58 views

Simulation of the cdf with copulas [duplicate]

Is this a proper way to simulate the joint cdf of normal rvs with perfect positive correlation? I followed these steps: I generate 10 000 observations for two independent uniform rvs, $U_1\sim U(0,...
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Copula vs conditional distribution

Let $\xi_1,\xi_2$ be two real-valued random variables with joint distribution $P$. The latter gives us marginals $p_1,p_2$ conditional distribution $p_{12},p_{21}$ and the copula $C$. The latter ...
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What does a copula density explain about dependence of random variables?

I am studying copulas and I find it difficult to understand what a copula density tells me about the dependence of random variables. For example, if I have a Gaussian copula density, what can I say ...
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299 views

Appropriate number of degrees of freedom in t-Copula

In a consultation paper (EBA/CP/2014/08) the European Banking Authority (EBA) wrote: “it is proposed […] that Gaussian or Normal like Copulas are not to be used for operational risk modelling. For ...
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81 views

Simple Student Copula simulation

I want to simulate a t copula with given correlation parameter $\Sigma$ and $k$ degrees of freedom. I can't find any literature about practical simulation, so I am trying new approaches. (I also have ...
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44 views

showing that a function is a copula

In general, is there an easier way of showing that a function is a copula than showing that: $C(u_1,\dots,u_d) =P(U_1\le u_1,\dots,U_d \le u_d) \quad$is nondecreasing in each $u_i \in [0,1] $ $C(1,...
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Does principal components analysis lose any information regarding the interdependence of the variables?

I have often heard that a copula describes in full the interdependence of a set a random variables. Lets say I want to generate a set of random variables that conform to an observed joint probability ...
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Dependent identically distributed random variables

If $X_1$ and $X_2$ are dependent identically distributed, can we show that $Pr(X_1>X_2) = Pr(X_2>X_1)$? For i.i.d, it is obvious, but what if they are dependent?
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survival function in terms of copula

Is the survival copula the equivalent of the multivariate survival function? In other words, can I write $\bar{C}(u,v) = S(u,v) = u+v-1+C(1-u,1-v) $ where $\bar{C}$ is the survival copula, $S$ the ...
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135 views

How to choose what copula to use for a certain application?

I'm using the copula package in R for modelling dependece using copulas. 1)What is the suggested course of action for choosing a copula model? 2)Should I use the function ...
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374 views

Farlie-Gumbel-Morgenstern copula

I have the Farlie-Gumbel-Morgenstern copula and I want to generate two gamma marginals and find an expression for the linear correlation. I understand that to get the random variates $(u,v)$ I need to ...
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81 views

Copula Calibration

I've developed a step by step procedure for estimating a copula based upon 2 stock time series returns but I don't understand and have not implemented one step that is discussed in most of the copula ...
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21 views

linear correlation formula and maximum using Copulas

I am currently working on linear correlation between two random variables functions (say cdf $F$ and $G$). As far as I understand it is possible to correlate those random variables with copulas. For ...
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129 views

Simulating from empirical copula density estimate

I went through these excellent slides by Prof. Arthur Charpentier on the use of Kernel density estimators (KDEs) with boundary correction to estimate Copulas in a nonparametric fashion. As explained ...
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Measuring dependence of variables with measurement errors and copulas

The weight of 100 subjects is measured twice resulting in the normally distributed $X_1\sim N(\mu_1,\sigma_1)$ and $X_2\sim N(\mu_2,\sigma_2) $. I am interested in the correlation between the ...
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55 views

Analytical Formula for Exceedance Correlation of Multivariate Normal Distribution and others

I understand that the Exceedance Correlation concept. Knowing that the multivariate normal distribution has asymptotic independence on the tail, lower and lower exceedance correlation approaching to 0 ...
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71 views

Help understanding copula version of Spearman rank correlation

I'm reading through this article (http://www.sciencedirect.com/science/article/pii/S0047259X06000662) where they have a population version of the Spearman rank correlation. I'm having a little bit of ...
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Why is sklar's theorem written for bivariate case only?

I read An Introduction to Copulas by Nelsen. Just wonder, Why is sklar's theorem written for bivariate case only?
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How do I implement a copula to transform a multivariate normal distribution to handle dichotomous variables?

I have a bivariate normal distribution where both dependent variables are dichomotous. I can estimate this fine in R using the multivariate normal distribution, but how can I implement a copula to ...
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1answer
39 views

How to efficiently simulate values from a multivariate normal given one of the components?

Suppose $X, Y_i$ for $i=1...n$ are standard normal variable but are also correlated so collectively they come from a multivariate normal distribution. Now the complication is what if I want to ...
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116 views

How to find copula-based conditional probability P(U|V>=v)?

Using the Copula operator $C$, which for any (possibly dependent) RVs $U$ and $V$ represents the joint cumulative DF of their inverse probability transform. That is, $U^* = F^{-1}_U (U) \sim \mbox{...
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61 views

Spearman's correlation as a parameter

Spearman's rank correlation for a bivariate sample $\{ (x_1, y_1), (x_2, y_2) , \ldots , (x_n, y_n) \}$ is generally defined as the correlation between the ranks of the observations, but what is the ...
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50 views

Fitting a Copula to Two Stochastically Dependent Variables in R

I have two sets of observed data, and I would like to model their joint distribution using a copula in R. I have transformed each set into a uniform distribution using their CDFs, and so now I have ...
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55 views

Comonotonic and Countermonotonic RV's and their relation to Frechet Hoeffding Bounds

If $F_1 ... F_d$ are are all continuous, and $X_j \sim F_j$, $j= 1...d$, then the Frechet upper bound corresponds to comonotonic random variables with $$ X_j = F^{-1}_j(F_1(X_1)) \ \ \ \ \ \ \ \ [1] $...