# 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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### Is it possible to have a pair of Gaussian random variables for which the joint distribution is not Gaussian?

Somebody asked me this question in a job interview and I replied that their joint distribution is always Gaussian. I thought that I can always write a bivariate Gaussian with their means and variance ...
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### Conditional expectation of two identical marginal normal random variables

Let $Y_0$ and $Y_1$ be both identically (not necessarily independent) normally distributed with mean $\mu$ and $\sigma^2$, i.e., $Y_i \sim N(\mu, \sigma^2)$ for $i = 1, 2$. Let $\rho$ denote the ...
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### 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 joint distribution using a Gaussian copula, denoted $C(F,G;\Sigma)$. All the ...
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### Attainable correlations for lognormal random variables

Consider the lognormal random variables $X_1$ and $X_2$ with $\log(X_1)\sim \mathcal{N}(0,1)$, and $\log(X_2)\sim \mathcal{N}(0,\sigma^2)$. I'm trying to calculate $\rho_{\max}$ and $\rho_{\min}$ for ...
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For some time now, I have been looking for a good introductory reading on Copulas for my seminar. I am finding lots of material that talk about theoretical aspects, which is good, but before I move ...
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### How do I convert simulated values from a copula to "real" observations? R

I have managed to fit different kinds of copulas to my data in R (mostly Archimedean copulas) using the copula package. I have no problem simulating pseudo observations (u and v), my questions are: ...
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### Derivation of survival copula

\begin{align}F(x,y) &= P(X\leq x, Y\leq y) \\ &= C(F(x), F(y)) \end{align} The copula of a bivariate distribution is equal to its CDF, $C(u,v) = F(x,y)$. The density of the data's CDF ...
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### Difference between multivariate standard normal distribution and Gaussian copula

I wonder what the difference between multivariate standard normal distribution and Gaussian copula is since when I look at the density function they seem the same to me. My issue is why the Gaussian ...
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### Spearman $\rho$ as a function of Pearson $r$

It is common to talk about the linear correlation, Pearson's $r$, between two random variables $\{(x_1,y_1),(x_2,y_2),\ldots,(x_n,y_n)\}$ as having two components: a) the copula and b) the marginal ...
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### 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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### Copulas with Regression

Copulas are joint distribution of uniform marginal distributions. Traditionally I have seen examples of fitting a Copula to the data and then simulating from the data. I haven't seen much on Copula ...
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### Proof of the relation between Kendall's Tau and Pearson's Rho for the Gaussian Copula

I know in the case of the bivariate normal distribution Kendall's Tau is given by $$\tau=\frac{2}{\pi}\arcsin({\rho})$$ where $\rho$ is Pearson's correlation. Can someone given a derivation of this ...
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### How to forecast from GARCH-copula model?

I am reading to understand how to forecasting time-series data from the GARCH-copula model. I am looking forward to understanding the steps. From my understanding, we should follow the following steps:...
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### Dependent thinning Poisson process

If $N_1$ and $N_2$ are independent Poisson processes then the superposition is a Poisson process. Is it possible to construct two dependent Poisson processes such that the superposition is a Poisson ...
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### Constructing a joint distribution from pairwise bivariate marginal distributions?

It's fairly well-known that given univariate distribution functions $F_X, F_Y, F_Z$, one can construct the joint distribution $F_{(X, Y, Z)}(x, y, z) = C(F_{X}(x), F_{Y}(y), F_{Z}(z))$, where $C$ is ...
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### Generating values from copula using copula package in R

I have a bunch of questions concerning the use of the copula package in R. My overall aim is to generate synthetic values using copulas. I am analyzing a hydrological data: annual peak discharge [m³/s]...
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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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### Sampling from copula given a particular value of marginal

I am working in the bivariate case with copulas as follows: I have two marginal Gamma distributions $f_1=Ga(\alpha_1, 1)$ and $f_2=Ga(\alpha_2, 1)$ that are bound by a copula Frank copula $C$, with ...
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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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### What is the expectation of the following joint CDF?

Yesterday, I asked the following question regarding copulas: "Let's say $X=(X_1,X_2)′$, where $X\in \mathbb R^2$. What is the expectation of the copula function $C(F_{X_1}(x_1),F_{X_2}(x_2))$ - i.e....
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### Correlated Bernoulli trials, multivariate Bernoulli distribution?

I'm simplifying a research question that I have at work. Imagine that I have 5 coins and let's call heads a success. These are VERY biased coins with probability of success p=0.1. Now, if the coins ...
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### How to fit a copula when zeros abound?

I am modelling a joint distribution for two random variables: $F(x,y)$. I observe $n$ data points $(x^{}_{i},y^{}_{i})^{N}_{i=1}$. I would like to model $F$ as the product of its marginals and a ...
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