26
votes
Accepted
Correlated Bernoulli trials, multivariate Bernoulli distribution?
No, this is impossible whenever you have three or more coins.
The case of two coins
Let us first see why it works for two coins as this provides some intuition about what breaks down in the case of ...
- 378
20
votes
Why don't we see Copula Models as much as Regression Models?
The first and most important reason is that standard regression models had a one to two-hundred year headstart on copula models (depending on exactly where you count the genesis of regression models ...
- 130k
14
votes
Why don't we see Copula Models as much as Regression Models?
A reason might be that regression and copulas do not answer the same question. Copulas are about the joint distribution while regression is about a conditional distribution or just the conditional ...
- 68.9k
13
votes
Spearman $\rho$ as a function of Pearson $r$
I think I found the answer. In Pearson's "On further methods of determining correlation" (1907) he derives the expression:
$$
r=2 \sin \Big(\frac{\pi}{6}\rho\Big),
$$
which implies,
$$
\rho= \frac{6}{\...
- 271
11
votes
How to construct a multivariate Beta distribution?
It is natural to use a Gaussian copula for this construction. This amounts to transforming the marginal distributions of a $d$-dimensional Gaussian random variable into specified Beta marginals. The ...
- 330k
11
votes
Why don't we see Copula Models as much as Regression Models?
A short answer is that in practice for many applications we don't need the joint probability distributions. A cynic would say that it's also because the users don't event understand what is a joint ...
- 61.9k
10
votes
Accepted
Quantifying dependence of Cauchy random variables
Just because they don't have a covariance doesn't mean that the basic $x^t\Sigma^{-1} x$ structure usually associated with covariances can't be used. In fact, the multivariate ($k$-dimensional) ...
- 40.7k
9
votes
How to make random draws from an unspecified distribution?
Presumably $w_1+w_2=1$ and $w_1,w_2 \geq 0$, so $f$ is a convex combination of $f_1,f_2$ and therefore a valid distribution (a mixture of $f_1,f_2$).
Generate a Bernoulli($w_1$) random variable (i.e. ...
- 636
9
votes
Why is Gaussian Copula's Tail Dependence Zero?
For a non-technical, intuitive view of what the tail index is telling you, we can look at simulation and compute sample estimates of the quantity $P[F(Y) > q | F(X) > q]$ as $q$ increases.
Here ...
- 287k
9
votes
Accepted
Expressing a marginal probability using copulas
The issue of notation seems crucial. I propose, therefore, to disambiguate the ubiquitous and overloaded "$f$" by means of subscripts. Thus, $f_{XYZ}$ will be the full density function and ...
- 330k
9
votes
Accepted
Can every multivariate distribution be expressed as a function of univariate distributions of the same random variables?
YES
This is Sklar's theorem.
Let $H(x_1,\cdots,x_d) = \mathbb P\big(X_1 < x_1,\cdots,X_d<x_d\big)$ be a multivariate CDF, and let $F_i$ be the marginal CDFs. Then there is a function, $C$, ...
- 65.8k
8
votes
Accepted
Does assumption of normality of each mixture components implies that each margins is normal
Multivariate Gaussian mixtures are not themselves multivariate Gaussian, their components are.
Statements that apply to the components of a mixture don't generally apply to the mixture (this would ...
- 287k
8
votes
Quantifying dependence of Cauchy random variables
While $\text{cov}(X,Y)$ does not exist, for a pair of variates with Cauchy marginals, $\text{cov}(\Phi(X),\Phi(Y))$ does exist for, e.g., bounded functions $\Phi(\cdot)$. Actually, the notion of ...
- 107k
8
votes
Accepted
How to forecast from GARCH-copula model?
How to fit a copula GARCH model?
For each series (margins):
(a) fit a univariate GARCH model (e.g. using ugarchspec followed by ...
- 68.9k
8
votes
How can I generate random observations from a concrete copula?
Copulas are usually defined via the joint cdf of the Uniform components,
$$ C(u_1,u_2,\dots,u_d)=\mathbb P[U_1\leq u_1,U_2\leq u_2,\dots,U_d\leq u_d]$$
Unfortunately, a value $C(u_1,u_2,\dots,u_d)$ ...
- 107k
8
votes
What is the formula for the conditional inverse function for the Ali-Mikhail-Haq and the Farlie-Gumbel-Morgenstern Copulas?
By definition, when $(U,V)$ has a copula distribution $C,$
$$C(u,v) = \Pr(U\le u,\ V \le v).$$
To find the distribution conditional on $u$ when $C$ is differentiable at $u$ with derivative $C_u(u,v)$ ...
- 330k
7
votes
Accepted
Does anyone know what the intuition of H-volume is?
The H-Volume is the volume contained by the rectangle $[x_1,x_2] \times [y_1,y_2]$ of a 3-dimensional function $H(x,y)$. To visualize this, see the Figure
which is the contour plot of the ...
- 872
7
votes
Accepted
Conditional expectation of two identical marginal normal random variables
I don't think the formula given in the question can be correct in all cases, it is developed using joint normality. Without joint normality we can use copulas. For $X,Y$ random variables with joint ...
- 81.7k
7
votes
Is there a bivariate $\beta$ distribution I can fit to my data?
There are many ways to define bivariate beta distributions, that is, bivariate distributions on the square $[0,1]\times [0,1]$ with beta marginals.One way is to start with the usual stochastic ...
- 81.7k
7
votes
Accepted
When modeling a copula, you need to generate "pseudo observations"? Why? What is a pseudo observation?
Copula models based on pseudo-observation (normalized ranked data), not on the original dataset. That due to the Sklar's theorem (the backbone of the copula model). From Sklar's theorem, copula is a ...
- 1,660
7
votes
Accepted
What will Frank copula tell me?
More generally, how do we choose which copula model to use in a given problem? The main guiding principle I learned is to choose a copula model based on the dependence structure of the variables.
...
- 5,037
7
votes
How can I generate random observations from a concrete copula?
A recipe for how to generate random observations depends on the level of detail. If you can generate from the copula, then you get points on $[0,1]\times [0,1]$. Then you transform the $X$ coordinate ...
- 68.9k
7
votes
How to write a function for the normal copula in R?
By definition, the bivariate CDF for variables $(X,Y)$ is
$$\Phi_\theta(x,y) = \Pr(X\le x,\ Y\le y).$$
It is the area of an infinite rectangle (with upper right vertex at $(x,y)$) weighted by the ...
- 330k
7
votes
Accepted
Why does the multivariate data generated by a copula in R not exhibit the prespecified correlation?
Three things to note:
Pearson correlation is not preserved through strictly monotonic transformation of the margins, including marginal transformations used with copulas, except in special cases. ...
- 287k
6
votes
Accepted
An example of a bivariate pdf, where marginals are triangular distributions
A nice way to do this is to use copulae.
In your case:
let $X \sim \text{Triangular}(0,1)$ with pdf $f(x)$ and parameter $b$, and
let $Y \sim \text{Triangular}(0,1)$ with pdf $g(y)$ and parameter $...
- 7,931
6
votes
Accepted
What is the advantage of modelling dependency using Copulas?
Before I write the answer to my question, I would like to post some excellent answers already given for a similar question
First timers, usually, do not appreciate the modeling flexibility associated ...
- 1,572
6
votes
Accepted
what does positive Gaussian copula dependency describe
How rigorous do you want the exposition to be? In simple terms, even though we often talk about "dependence" and "correlation" in intuitive terms, they have formal definitions. In ...
- 5,037
6
votes
What is the copula of a variable with itself?
I see two issues with your question:
You need to define more rigorously what you mean by "$X$'s joint distribution with itself"
I think it is confusing or maybe even inappropriate to use ...
- 2,763
6
votes
Independent copula vs Student-$t$ copula with zero correlation matrix?
The uncorrelated $t$ copula is not the same as the independence copula. It is based on the multivariate $t$-distribution, which is an elliptical family, and the only elliptical distribution for which ...
- 81.7k
6
votes
Bivariate Distribution with Uniform Marginals is Bound to be Uniform?
No, the joint distribution is not necessarily uniform.
Consider $X$ and $Y$ with a joint pdf
$$
f(x,y) = \begin{cases}
2, \text{if } x \in (0,0.5), y \in (0,0.5)\\
2, \text{if }x \in (0.5,1), y \in (0....
- 2,700
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