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Suppose the marginal distributions of $X$ and $Y$ are both $U[a,b]$, where $U[a,b]$ denotes uniform distribution on $[a,b]$. When $X$ and $Y$ are correlated, how to write their joint density for some prespecified measure of correlation?

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    $\begingroup$ In case it comes up, and I think it will (or at least should), do you know the term “copula”? Also, do you have a plot in mind? It would help to have some idea of what you mean by correlation. For instance, according to you, for $X \sim U(0,1)$, are $X$ and $X^2$ correlated even though the relationship isn’t linear? (Remember that for this $X$, $cor(X,X^2)\ne 0$.) $\endgroup$
    – Dave
    Commented May 30, 2020 at 3:57
  • $\begingroup$ @Dave Thanks. Yes, the correlation is not restricted to be linear. I guess indeed copula need to be used here. $\endgroup$ Commented May 30, 2020 at 5:25
  • $\begingroup$ One possibility, with positive correlation is as follows: $f(x,y) = 2$ if $0<x<\frac12, 0<y<\frac12$ or $\frac12 < x <1, \frac 12 < y <1$, and 0 otherwise. Another with positive correlation has $X\equiv Y,$ concentrating all probability on a line. Similarly, $X\equiv -Y$ has negative correlation. $\endgroup$
    – BruceET
    Commented May 30, 2020 at 6:49
  • $\begingroup$ @BruceET in your first example, is that just filling in the first and third quadrants (so to speak)? $\endgroup$
    – Dave
    Commented May 30, 2020 at 7:07
  • $\begingroup$ @Dave: That was my intention; never quite sure how JaX works in Comment mode. $\endgroup$
    – BruceET
    Commented May 30, 2020 at 7:27

1 Answer 1

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I think the right place to start is what correlation means in a nicer setting: Gaussian, of course.

In the bivariate Gaussian distribution, the population correlation is given by the $\rho$ parameter. This tells us something about the fit to a line.

I have tried fitting marginal uniform distributions to a line and failed, but we can still get at the $\rho$ parameter of the bivariate Gaussian.

That $\rho$ parameter is the parameter of the bivariate Gaussian copula, and we can use a Gaussian copula with more than just Gaussian marginal distributions. We can use uniform marginal distributions and specify the parameter in the Gaussian copula, giving us a way to describe the relationship between the two marginal distributions.

When I have simulated this, I have not gotten the sample correlation to be the specified parameter of the Gaussian copula, so I do not know if this quite counts as specifying the correlation. However, the estimate in R's cor function is biased, perhaps accounting for this discrepency. I welcome discussion in the comments about this topic.

(Remember that a biased estimator isn't inherently a bad estimator; in fact, $s$ is a biased estimator for standard deviation, even though $s^2$ is unbiased for variance, so we use biased estimators all the time.)

library(copula)
set.seed(2020)
N <- 1000
R <- 10000

# define the copula
#
nc <- normalCopula(param = 0.81)

# Define the population distribution with the nc copula and U(0,2) marginals
# 
unif_unif <- mvdc(nc, c("unif","unif"),list(list(min=0, max=2),list(min=0, max=2)))
v <- rep(NA, R)
for (i in 1:R){

  # Sample from the population
  #
  D_uu <- rMvdc(N, unif_unif)

  # Calculate the correlation
  #
  v[i] <- cor(D_uu[,1], D_uu[,2])
  if (i %% 250 == 0){print(i)}
}
plot(density(v))
abline(v=0.81)
mean(v) # 0.7963396, slightly lower than the specified 0.81

EDIT

Let's look at some pictures. Here is the last distribution produced in the simulation: plot(D_uu).

enter image description here

Now compare to a bivariate normal distribution with population correlation $0.81$: plot(qnorm(D_uu[, 1]/2), qnorm(D_uu[, 2]/2)). (This uses the same (Gaussian) copula to relate the marginals but gives each marginal a normal distribution.)

enter image description here

Both plots have the data kind of hugging the line $y=x$, and both see a bulge in the perpendicular direction. However, the behavior of the marginal normal distributions causes more points to be near the mean than in the marginal uniform distributions (consider the PDFs).

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