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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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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32 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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14 views

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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92 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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19 views

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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53 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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28 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
49 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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48 views

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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46 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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52 views

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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48 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 ...
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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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123 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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54 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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41 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] $ ...
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1answer
80 views

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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179 views

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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1answer
80 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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345 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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66 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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19 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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118 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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41 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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56 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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37 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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100 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 ...
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56 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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44 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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52 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] ...
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108 views

Simulating random variables given partial distributions and correlation

After Monte Carlo simulations I obtained approximated distributions for X and Y. Now I want to add some form of correlation between them. To simulate random variables from a distribution the idea is ...
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1answer
80 views

Copula density function

In the equation $h_t(x,y|...) = ...$, can anyone explain me why the first derivatives of the marginal distributions are included? $H_t$ is a distribution function and $h_t$ its density function. ...
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Assessing dependencies

I am looking for a method to assess the dependencies between random variables (cross sectional data). I know that I can use copulas models, but I'd like to see if there are any other alternatives.
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What is the equivalent for cdfs of MCMC for pdfs?

In conjunction with a Cross Validated question on simulating from a specific copula, that is, a multivariate cdf $C(u_1,\ldots,u_k)$ defined on $[0,1]^k$, I started wondering about the larger picture, ...
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Issues with derivative of a copula density

Suppose I have a bivariate copula density with marginals $F(X)$ ($F(X)$ is normal with parameters $\mu_1$, $\sigma_1^2$) and $G(Y)$ ($G(Y)$ is normal with parameters $\mu_2$ and $\sigma_2^2$). I want ...
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How to simulate from a log-copula function?

Does anybody know how to simulate from a log-copula function? I'm trying to simulate $(u,v)$ from a log-copula function with the CDF: $$ C(u,v, a) = \exp\bigg(1-\big[(1 - \ln u)^a + (1 - \ln v)^a - ...
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Bayesian inference on default probabilities

I have a doubt, I don't have great experience in Bayesian inference and I wondered if it is possible to construct the following model: I'm interesting in copula-dependent probabilities of default ...
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Why BiCopDeriv function in VineCopula package creates NaNs/Inf?

My question is why BiCopDeriv and BiCopDeriv2 function in Vine Copula package of R which produces first and second derivatives of copula density produce NaNs/Inf in this example. Attached is the code. ...