# Extending the Hellinger Distance to multivariate distributions

The hellinger distance for a univariate distribution is

$$\ H(x) = 1 - \int \sqrt {f(x)g(x)}\; dx$$

I wish to use it for a bivariate distribution, by extending it to this form

$$\ H(x) = 1 - \int\sqrt {f(x, y)g(x, y)} \;dx dy$$

Can someone comment on whether this is still a valid Hellinger distance?

• By "$H(x)$" do you perhaps mean $H(f,g)$? If not, then what is "$x$"? BTW, it seems to me that the definition of Hellinger distance makes no reference to the dimension. – whuber Mar 20 '13 at 15:27
• Oops, you're right it should be H(f,g) – gamerx Mar 20 '13 at 16:31

It's definitely valid. In fact, Hellinger distance between two measure $P$ and $Q$, which are dominated by the same (sigma-finite) measure $\lambda$, is defined as $$H^2(P,Q) = \frac{1}{2}\int\left( \sqrt{\frac{dP}{d\lambda}} - \sqrt{\frac{dQ}{d\lambda}} \right)^2 \, d\lambda \, .$$ Therefore, you're considering the special case when $\lambda$ is two-dimensional Lebesgue measure, and the densities $f$ and $g$ are versions of the Radon-Nikodym derivatives $dP/d\lambda$ and $dQ/d\lambda$.