# Mathematical Statistics – Are $U=\frac{2X_1^2}{(X_2+X_3)^2}$ and $V=\frac{2(X_2-X_3)^2}{2X_1^2+(X_2+X_3)^2}$ Independent?

independencemathematical-statisticsnormal distributionself-study

Consider i.i.d standard normal variables $$X_1,X_2,X_3$$. How can I determine whether $$U=\frac{2X_1^2}{(X_2+X_3)^2}$$ and $$V=\frac{2(X_2-X_3)^2}{2X_1^2+(X_2+X_3)^2}$$ are independently distributed?

This was part of a multiple choice question, so I am wondering if there is a short argument. I can show that $$Y_1=X_1,Y_2=X_2-X_3,Y_3=X_2+X_3$$ are all independent of each other. And $$U$$ is a function of $$(Y_1,Y_3)$$ while $$V$$ is a function of $$(Y_1,Y_2,Y_3)$$. Also, $$U$$ and $$V$$ are functionally dependent since $$V=\frac{2Y_2^2/Y_3^2}{U+1}$$. But that doesn't help me answer the question. It can be seen that the marginals of $$U,V$$ are $$F$$ distributions. Do I have to find the joint density of $$(U,V)$$ through a change of variables?

Another idea was to try applying Basu's theorem. So I introduced a parameter $$\sigma^2$$ as the variance of the $$X_i$$'s. But then both $$U$$ and $$V$$ seem to be an ancillary statistic for $$\sigma^2$$.

Let $$X=X_1,$$ $$Y=(X_2+X_3)/\sqrt2,$$ and $$Z=(X_2-X_3)/\sqrt2.$$ As in your question, it is apparent that these are iid standard Normal. Moreover,

$$U = \frac{X^2}{Y^2}\ \text{ and }\ V = 2\frac{Z^2}{X^2 + Y^2}.$$

Consider, then, how $$X/Y$$ and $$X^2+Y^2$$ are related. The latter is the squared distance from the origin while the former is the cotangent of the angle $$\Theta$$ made by $$(X,Y)$$ to the $$X$$-axis. Because iid Normal variables are spherical, the angle and the distance are independent.

Recall that any (measurable) functions of independent variables are independent. Call these functions $$f$$ and $$g.$$ Beginning with the variables $$\Theta$$ and $$(R, Z) = (\sqrt{X^2+Y^2}, Z),$$ we have just seen $$(\Theta,R,Z)$$ are independent. Observing that we may express $$V = 2Z^2/R^2 = g(R,Z)$$ and $$U = \cot^2(\Theta) = f(\Theta),$$ we immediately see $$(U,V)$$ are independent.

In retrospect, it is evident $$(\Theta, R, Z)$$ is a cylindrical coordinate system for the original Cartesian coordinates $$(X_1,X_2,X_3).$$ Thus, if you prefer an explicitly rigorous, Calculus-based derivation, consider computing the joint distribution function in these cylindrical coordinates: it should separate into a term for $$\theta$$ and a term for $$(r,z).$$

BTW, this is a challenge for those of us who like to draw inspiration from simulations: both $$U$$ and $$V$$ have infinite means and, in even fairly large simulations, the $$(U,V)$$ scatterplot can look decidedly dependent. Here, for instance, is a log-log plot for 4,000 random $$(U,V)$$ pairs.

The lack of many extreme values of $$U$$ makes it look like $$V$$ tends to be high when $$U$$ is extreme.

Here's the R simulation code.

n <- 4e3
x <- rnorm(n)
y <- rnorm(n)
z <- rnorm(n)

u <- 2*x^2 / (y+z)^2
v <- 2*(y-z)^2 / (2*z^2 + (y+z)^2)
plot(u,v, log="xy", col=gray(0, alpha=0.1), asp=1)