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What could be a form of $$f_{X,Y}(x,y)$$ where $f_X(x)$ and $f_Y(x)$ both have the form of a triangular distribution with support $(0,1)$, but with different parameters that governs location of mode?

Some discussion on it could be found in

Eagleson, G.K., Lancaster, H.O.: The regression system of sums with random elements in common. Australian Journal of Statistics 9, 119–125 (1967)

and

Balakrishnan, N, Lai, Chin-Diew: Continuous Bivariate Distributions (2009)

kjetil b halvorsen
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Sergey
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1 Answers1

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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 $c$:

enter image description here

with cdf's $F(x)$ and $G(y)$:

enter image description here

... where I am using the Prob function (from the mathStatica package for Mathematica) to automate the nitty-gritties of the cdf calculation.

Then, define a copula function, which is a function of the two cdf's $F$ and $G$ that creates a bivariate joint distribution function (cdf) from $F$ and $G$, such that the marginal pdf's of $X$ and $Y$ are still $f$ and $g$ respectively. Here, I use a Morgenstern copula with parameter $\alpha$ that induces correlation (there are many many other Copula functions available):

enter image description here

Let $h(x,y)$ denote the bivariate Triangular joint pdf obtained via a Morgenstern copula. Here we differentiate the Copula function (joint cdf) to derive the joint pdf $h(x,y)$:

enter image description here

The following diagram plots the joint pdf $h(x,y)$ when $b = \frac12$, $c = \frac34$ and $\alpha = 0$ (independent):

enter image description here

Here is the same plot when $\alpha = -1$:

enter image description here

wolfies
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    Omg. Thank you so much. I knew that I need to do copula and tried to do something myself but then I thought that I'm likely to make a mistake and decided to go with a uniform density for both RW This stuff is so helpful. I'll use exactly this in the paper I'm doing. Thanks a lot!!! – Sergey Apr 24 '18 at 09:40
  • Hey Wolfies, I can not replicate your results. For some reasons, Mathematica does not return Prob function properly... Any chance you could explain a little a tiny bit further, maybe you skipped something? Also, I have the latest version, could there be some incompatibilities? – Sergey May 01 '18 at 07:44
  • Nope, nothing works, embarrassing. Is there any chance you could post your *.nb? Is that something people do here? – Sergey May 01 '18 at 07:59
  • @Sergey Note that the `Prob` function belongs to the _mathStatica_ package, which is an add-on to _Mathematica_. – wolfies May 01 '18 at 08:08