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Questions tagged [copula]

A copula is a multivariate cumulative distribution function for which the marginal probability distribution of each variable is uniform on the unit interval.

A copula is a multivariate cumulative distribution function for which the marginal probability distribution of each variable is uniform on the interval $[0, 1]$. Copulas are used to describe the dependence between random variables. Copulas have been used widely in quantitative finance to model and minimize tail risk and portfolio-optimization applications.

Sklar's theorem states that any multivariate joint distribution can be written in terms of univariate marginal distribution functions and a copula which describes the dependence structure between the variables.

Copulas are popular in high-dimensional statistical applications as they allow one to easily model and estimate the distribution of random vectors by estimating marginals and copulae separately. There are many parametric copula families available, which usually have parameters that control the strength of dependence.

Two-dimensional copulas are known in some other areas of mathematics under the name permutons and doubly-stochastic measures.

Source: Wikipedia

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Proof of Frechet-Hoeffding Copula bounds

How is the lower Frechet-Hoeffding copula bound proved? In the bivariate case, it follows from $C(u_1,u_2)-C(u_1,v_2)-C(v_1,u_2)+C(v_1,v_2)\geq0$ by setting $(v_1,v_2)=(1,1)$. I'm struggling to prove it in higher dimensions.
user62764
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Product of correlated random variables and its transformation

There is an interesting result, saying that if $Z_1, Z_2$ are standard normal random variables with a correlation $\rho\in (-1,1)$, then the product $Z=Z_1Z_2$ has a density function explicitly given…
Albert Paradek
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Proving that copulas are Lipschitz continuous

A copula is a function $C:[0,1]^2\to[0,1]$ such that $C(x,0)=C(0,x)=0$ for all $x\in[0,1]$, $C(x,1)=C(1,x)=x$ for all $x\in[0,1]$, and \begin{equation}\label{ineq} C(x_2,y_2)-C(x_1,y_2)-C(x_2,y_1)+C(x_1,y_1)\ge0\tag{*} \end{equation} for all…
Cm7F7Bb
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How to derive the density of Gaussian Copula?

I have a question regarding Gaussian copulas: The multivariate Gaussian copula is defined as, $$ C(u_1,\dots,u_n;\Sigma) = \Phi_{\Sigma}(\Phi^{-1}(u_1),\dots,\Phi^{-1}(u_n)), $$ where $\Phi_{\Sigma}$ is a multivariate $n$-dimensional normal…
Wawel100
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How do we prove that $\max\{x_1 + x_2+ \ldots + x_n - n + 1,0\} \leq C(\textbf{x}) \leq \min\{x_1,x_2,\ldots,x_n\}$?

I am interested in proving the generalized version of the Fréchet-Hoeffding inequality. Precisely speaking, given a $n$-copula $C:[0,1]^{n}\rightarrow[0,1]$, how do we demonstrate that $$ \max\{x_1 + x_2 + \ldots + x_n - n + 1, 0\} \leq…
user0102
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How do I derive a pair-copula decomposition for a joint density function?

In Section 4.1 of Analyzing Dependent Data with Vine Copulas (Czado), the author decomposes a three-dimensional joint density function into bivariate copula densities and marginal density functions. I’m having trouble with this derivation and would…
Leon
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Derivation of bivariate Gaussian copula density

The multivariate Gaussian copula density, derived here, is $$c(u_1,\ldots,u_n;\Sigma)=|\Sigma|^{-\frac{1}{2}}\exp\!\left(-\frac{1}{2}x^{\top}(\Sigma^{-1}-I)x\right)$$ where $\Sigma$ is the covariance matrix, and …
develarist
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How to simplify function to $\log (1-\theta)+\frac{\theta}{2-\theta}$?

\begin{align} f(\theta)& = \log \frac{1}{1-\theta}-\frac{\theta}{2-\theta}+\frac{\theta^{2}}{(2-\theta)^{2}} \\ & = \log (1-\theta)+\frac{\theta}{2-\theta} \end{align} How are the two equations above equal to each other, i.e. reduce the first into…
develarist
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Simplify $\mathbb{E}\left[ -\log c(u,v)\right] $, the expected logarithm of the copula density

2020 11.26: I finished the derivation but haven't posted it here. I leave this open so that those willing to try their own version can if they want Question How can we derive a closed-form analytical solution for $\mathbb{E}\left[ -\log…
develarist
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Archimedean Clayton copula entropy

Question I would like to derive the entropy of Archimedean parametric copulas (Clayton, Frank, or Gumbel), focusing here on the Clayton copula. Link to similar question on the t-copula. The bivariate copula function, $C$, for the Clayton copula,…
develarist
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Empirical Copula when there are thresholds for no data pairs

I am trying to find the empirical copula linking two random variables $X$ and $Y$. I have some data available but it's limited with respect to the variable $Y$ and I am not convinced it's enough data and will lead to the right copula. The variable…
Math Girl
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Find $S$ such that $P(X+Y\in S)\geq 0.9$ only using marginal distributions of $X$ and $Y$

Let $X, Y$ be continuous random variables with their distributions $F_X, F_Y$, finite second moments and correlation $\rho$. I would like to find a smallest possible set $S$ (set such that its Lebesgue measure is as small as possible) satisfying…
Albert Paradek
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Example of convergent bivariate marginals, but joint distribution does not converge

I'm looking for an example of sequences of random variables $(X_n), (Y_n), (Z_n)$ such that $X_n, Y_n, Z_n \sim \mathcal U(0,1)$, $(X_n,Y_n), (X_n, Z_n), (Y_n, Z_n)$ converge in distribution, but the vector $(X_n,Y_n, Z_n)$ does not converge. Using…
Stefan Perko
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Variable change in Copula for the joint pdf of correlated random variables

Let $f_{X,Y}(x,y)$ be the joint probability density of correlated random variables $X$ and $Y$ based on a Copula $C$ (Gaussian in my case) where $f_X(x)$ and $f_Y(y)$ are the marginal probability density functions of $X$ and $Y$ respectively.…
Yakari Dubois
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What assumptions/properties, if any, characterize the Gaussian copula

I would like to know what assumptions and/or properties characterize the Gaussian copula. I am interested in the case of arbitrary dimensionality, and am not so interested in properties specific to the bivariate case. What do I mean by…
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