Interpretation of CopulaDistribution [closed]

Question

I was just wondering, when I call the CopulaDistribution function in Mathematica, am I calling its cumulative function or its density function?

I have looked up the help and am still a little bit unsure.

EDIT: In particular, what does it mean when I take a RandomVariate from this CopulaDistribution? Surely I would have to sample from either the CDF or PDF.

Answer 1

The question is of some interest because it captures rather nicely the difference between:

A. mathematical statistics ... where we work with characterisations of distributions, such as starting with a pdf, or cdf, or cf ... e.g. Let $X$ be a random variable with pdf $f(x)$:

$$f(x) = 1 -|x| \quad \text{for}\quad x\in(-1,1)$$

and

B. Mathematica's implementation of distributions ... which defines black box names to distributions that return nothing themselves ... but which we can then ask for a PDF or CDF or other characterisation from.

In this regard:

• The poster asks whether CopulaDistribution returns the pdf or the cdf? And the answer is neither.

• What does it return? Generally nothing.

Rather, CopulaDistribution .... like TransformedDistribution or MarginalDistribution, ... are Mathematica functions that don't actually seem to do anything. Do they involve any computation? No. Do they take up any processing cycles? No. They just return exactly what you enter e.g.

CopulaDistribution[{"FGM", .2}, {NormalDistribution[-1, 2], NormalDistribution[1, 1/2]}]

returns instantly:

CopulaDistribution[{"FGM", .2}, {NormalDistribution[-1, 2], NormalDistribution[1, 1/2]}]

Similarly:

TransformedDistribution[x^4, x \[Distributed] NormalDistribution[0, 1]

returns instantly the same input we entered ...

TransformedDistribution[x^4, x [Distributed] NormalDistribution[0, 1]

It doesn't even try to compute anything.

The only exception to this, that I am aware of, is where the solution is written up in advance ... much like a textbook appendix. So, for example:

TransformedDistribution[x^2, x \[Distributed] NormalDistribution[0, 1]]

ChiSquareDistribution[1]

... but no calculation or derivation is involved in this. It is just an appendix lookup (which is actually rather un-Mathematica-like, in my view).

RE comments: I don't know what is meant by a 'distribution is a distribution', and I think it is inherently wrong to suggest that distributions are black boxes, because it is not the way we tend to work or think about distributions in mathematical statistics. The starting point in mathematical statistics is not to define a black box, but to define a pdf (or a cdf or a cf). In effect, this is what Mathematica ultimately has to do anyway when we define our own custom density ... except that you have to manually create this artificial black box or placeholder, using:

dist = ProbabilityDistribution[1 - Abs[x], {x, -1, 1}]

ProbabilityDistribution[1 - Abs[x], {x, -1, 1}]

or CopulaDistribution or MarginalDistribution or ...

Once the black box is created, you can then 'operate' on it using PDF or CDF etc. The same goes for CopulaDistribution ... you have created a black box, and if you want something calculated, you will have to apply PDF or CDF etc to the latter.

Answer 2

I recently asked WRI about a similar behaviour on probability distributions. The answer was that the output IS generated, but erroneously the result IS NOT displayed.

Try

dist = ProbabilityDistribution[1 - Abs[x], {x, -1, 1}]

with

Mean[dist]

(* out 0 *)

or

cdist = CopulaDistribution[{"FGM", .2}, {NormalDistribution[-1, 2], 
NormalDistribution[1, 1/2]}]

with

Mean[cdist]
Variance[cdist]

(* out {-1, 1} and {4, 1/4} *)

So it does work. However, it is confusing.