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Questions tagged [copula]

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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differences between conditional probability and dependency

Sometimes, I read articles about conditional probabilities and other articles about conditional dependency. My question what is the main differences between them? For example, "https://en.wikipedia....
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VineCopula package conditional probability

Suppose I have 3 variables, $U_1, U_2, U_3$. Suppose further that I fit R or CVine copula. Then, to find the conditional probability $U_1 \leq u_1|U_2=u_2,U_3=u_3$, the package suggests using ...
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what does non-nested copulas mean?

I read from "Pair-copula constructions of multiple dependence", the following statement: "The likelihood of the Clayton copula is lower than that of the Student copula (39.72 vs. 47.81). However, ...
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How is this copula relation derived?

In the first answer to this question: Why is Gaussian Copula's Tail Dependence Zero? it says: Where does the second equality come from? It doesn't seem clear to me at all.
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Fitting wrong copula type to a real data set

I have developed a new mixture copula model. This model overcomes some limitation of copula models. I tested my new model on a simulation data. The model shows a superior result. My supervisor asked ...
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Quantifying dependence of Cauchy random variables

Given two Cauchy random variables $\theta_1 \sim \mathrm{Cauchy}(x_0^{(1)}, \gamma^{(1)})$ and $\theta_2 \sim \mathrm{Cauchy}(x_0^{(2)}, \gamma^{(2)})$. That are not independent. The dependence ...
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correlation measure in copula

I'm reading about copula but I'm a bit confused about it: we know that the Gaussian copula pdf is equal to : $$C (u_1, u_2, u_3,…, u_n; Σ) = |Σ|^{-1/2} \exp (-1/2 ε^{T}(Σ^{-1} – I)ε)$$ where $Σ$ is ...
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Best way to model the dependency of these two random variables (copula?)

I'm modelling the joint PDF of two variables that looks like this , where vt and vr are the random variables. The dashed line shows the joint pdf assuming they are independent (the product of its ...
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Finding expression of $n$-th derivative, when $n$ is large

For completeness, assume $C$ is an Archimedean copula with some generator function $\varphi$, which is usually assumed to have nice properties. It is known that $$ C(u_1, u_2, \ldots, u_n)=\varphi^{-1}...
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What is the preferred way to show that a 3-dimensional function is a copula?

I am currently working with 3 dimensions, and have some functions $C(u_1, u_2, u_3)$ for which I need to check whether or not they are a copula. What are all the requirements that I need to check? In ...
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What do the joint distributions look like? [duplicate]

I know that if I know the marginal distributions, that's not enough to specify the joint distribution. But obviously it can't be "any" joint distribution, it still needs to respect its marginal ...
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Is this exercise question well-posed? Dependence of marginal normals with correlation $\rho$

An exercise question asks me to find the tail-dependence of $X$ and $Y$, where $\operatorname{Corr}(X, Y) = \rho$, and both $X$ and $Y$ are standard normal variables. The tail-dependence is $$ \lim_{...
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How to calculate conditional probability when only marginals are known?

Say $X, Y$ are standard normals with correlation $\rho$. How do I calculate conditional probabilities such as $P(X \le x \mid Y \le y)$? I don't have any assumptions on their joint distribution, ...
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How to use multidimensional copula to obtain a joint distribution in python?

I am following this blog on how to use copula using python and scipy. From what I can understand, the process is as follows Generate samples from a multivariate distribution with a correlation (in ...
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sampling from multivariate distribution using copula

I'm trying to get a sample from a multivariate distribution which is constructed by copula. this is the steps that i go,is it true? at first i estimate the copula and the marginals then get a sample ...
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Does the empirical transformation of the margins will give copula data

I am learning coupla. I read that we need to transform the margins to the uniform distribution in order to get the copula data. Is that correct? Also, I read that to obtain copula data, we can ...
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Why does rank correlation only depend on copula, and not on the margins, if the margins are continuous

I read that for continuous margins, then Rank correlation only depend on copula and not on the margins. However, it is not the case for the non-continuous margins. Is that because the Kendall's tau (...
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Why aren't copulas unique for non-continuous random variables?

I have read from the Sklar's theorem that for continuous random variables, then copula is unique. I really do not understand why? Could someone please explain to me why the copula not unique if the ...
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Copulas with positive likelihood ratio dependence

Let $C(,)$ is an absolutely continuous two dimensional copula. Let the $C_{1}$, $C_{2}$, $C_{12}$ denote the derivatives wrt to the first, second, and mixed variables respectively. We say $C$ is PLR ...
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How to generate random variables which are correlated and yet marginally identically distributed? [closed]

I am wondering if there is a process to which we can generate random variables where there is a correlation structure between them, yet they are still marginally identically distributed? One idea that ...
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Why does correlation changes so much when converting data to pseudo observations for copula fitting?

I'm trying to model the joint distribution of time to failure of 2 (in future possibily more than 2) components. So far what I know is the following: The marginals of these components are ...
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Does this copula have a name?

Let \begin{equation} c(u_1,u_2|k) = k\,\big((1-u_1)\,(1-u_2)\big)^{k-1}\, _2F_1\!\left(1-k,1-k;1; \frac{u_1\,u_2}{(1-u_1)\,(1-u_2)}\right) , \end{equation} where $k \in \{1, 2, \ldots\}$ and where $...
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Proof for copula determined by correlation matrix

How can I proof that the copula of an elliptical distribution $El(\mu, \sigma^2, g_n)$ is fully determined by the generator function $g_n$ and the correlation matrix extracted from $\sigma^2$.
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Correct Interpretation of copula contour plots

Going into exploratory data analysis with the intention of fitting copula models, I was looking at the famous copula and they mention here that either contour or 3D ...
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Approximation of copulas

I'm studying copulas, finished the Introduction to Copulas by Nelsen. I'm interested in the latest/best known/etc approaches for approximating any Copula, or some families of copulas, so would be ...
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Default vs recovery rate - gaussian copula

I am trying to link defaut with recovery rates by using a Gaussian copula. My marginals follow a beta distribution. I would like to prove that the usage of a gaussian copula is good choice compared to ...
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Why do we use a criterion like AIC for Copula model selection?

If we look at the AIC formula: AIC = -2*log(ML) + 2k where k is the number of parameters in the model and is considered as the 'penalizing term' for complexity or over-fitting. Does this ...
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Skewed t copula model for multivariate

I recently try to write the code of multivariate skewed t copula model. but I do not have the right function of the model. may I ask for the right function or the matlab code of the model? thanks a ...
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Is there a reason most of author assumed i.i.d in copula models.

I read many of copula articles. I almost found that the authors assumed the i.i.d observations in copula models. I just wonder if there is a reason for this assumption? I searched this site and found ...
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Estimation of copula parameters for $C(u, v) = \min(u^a,v^b)\min(u^{1-a},v^{1-b})$

Given a bivariate copula, say $C(u, v) = \min(u^a,v^b)\min(u^{1-a},v^{1-b})$, $0 < a,b < 1$, how would we use the method of maximum likelihood to estimate the copula parameters, in this case $a$ ...
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Finding probability of a point using bivariate copula density

I have a data in the form $\textbf{N} \times 2$. I am using bivariate copula to model the joint density of this distribution. Firstly, I fit 2 marginal distributions independently on each column of ...
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Why do Gaussian copula does not have a closed form? hence, why numerical estimation is needed?

I am working on Gaussian copula. I always read that, Elliptical copulas do not have closed form expression and hence, the numerical estimation is needed. I really do not understand what does closed ...
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Calculate the implied correlation for missing cells in a correlation matrix in R

I have a correlation matrix in R. Many of the correlations are specified, but there are some that are "NA". eg, A __ B __ C A 100% NA 25% B NA 100% 50% C 25% 50% ...
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Questions about tail dependence of copula and copula parameters?

I would like to understand tail dependence and its relationship to the copula function. The relationship between copula and tail dependence can be expressed as: (from this question Understanding ...
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Trying to wrap my arms around copulas

This topic is dense with notation that makes things a bit confusing. But is this the correct interpretation? Suppose we have two jointly distributed random variables – $X$ and $Y$ – of arbitrary (...
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Fréchet Hoeffding bounds for symmetric random variables

(Edited to clarify the question). The Hakan & Demirtas (2012 doi: 10.1198/tast.2011.10090) approach to approximating Pearson correlation bounds uses the concept of the Fréchet-Hoeffding bounds by ...
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How do I sample arbitrary probability mass functions? (Archimedean Copula)

I'm trying to use an algorithm (Marshall, Olkin) for exchangable archimedean Copula to generate realizations of multivariate probability distributions. One step includes sampling V which is F ...
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How to derive a set of x and y lead to a given return period in multivariate distribution function

I constructed a multivariate distribution function using the copula package in R. I can successfully derive the cdf of a matrix of observational data. Now I want to find out the data associated with a ...
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Writing syntax for bivariate survival censored data to fit copula models in R

Am required to fit copula models in R for different copula classes particulary the Gaussian, FGM,Pluckett and possibly Frank (if i still have time). The data am using is Diabete data available in R ...
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Does assumption of normality of each mixture components implies that each margins is normal

I just would like to understand some information about the joint normality and the margins. I read that the normal joint distribution almost always implies that the univariate margins are all normal. ...
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Copula and non-Copula models

I am working with copula-based models. Copula models allow to models the margins separately from the dependencies structures. However, non-copula models do not allow for such separation. My question ...
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Does the distribution of each mixture components have any information of the margins of the variables? [closed]

I just start working with Gaussian mixture models and I just confused about some information, which I really would like to make sure that I understand the model very well. A Gaussian mixture model ...
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Simulation of t copula in Python

I am trying to simulate a t-copula using Python, but my code yields strange results (is not well-behaving): I followed the approach suggested by Demarta & McNeil (2004) in "The t Copula and ...
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Copulas with tail dependence only when correlation non-zero?

I'm wondering if I can get the best of both worlds of the Gaussian and T copulas (and if not, why not?). A property of the Gaussian copula that I like is that if the off-diagonal entry into the ...
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An example of a bivariate distribution with normal marginals and a nonlinear conditional mean curve? [duplicate]

That is, an example of a bivariate distribution with normal marginals for which a linear regression is inappropriate. Does an asymmetric copula always produce this?
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How to use Copulas to Combine Multivariate Conditional Probability with Univariate Conditional Probability?

This is sure to be an odd one, but here goes. I'm trying to estimate P(X|Y, Z) by the distributions of P(X|Y) and P(X|Z). I've thus far been trying to using copulas to achieve that aim, but I'm not ...
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When modeling a copula, you need to generate “pseudo observations”? Why? What is a pseudo observation? [closed]

I'm struggling with the concept of a "pseudo-observation." I can't find any material out there describing what it is in a simple, concise manner. Does it have something to do with observation's ...
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What is the differences between estimating the margins and transforming them using cumulative distribution function

In copula models, the estimation of copula parameters is based on the pseudo-observations of the original data. As I understand, we can transform the margins using the cumulative distribution function ...
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Probability with copulas

How can I express the conditional probability $P(X\leq x\mid Y\leq y)$ with copulas? I have seen, in another post, that you can do it, but with $P(X\leq x\mid Y=y).$
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How to add dependence between random vectors using a copula?

I understand that copulas can be used as a tool to add any conceivable dependence to a pair of random variables. However, I would like to add some dependence between two random vectors. Let us ...