Hellinger metrics of Weibull probability distributions The Hellinger distance between two probability distributions $f(x)$ and $g(x)$ is given by:
$$H(f,g)=1-\int_\Omega(dx\sqrt{f(x)g(x)})$$
My question is:
if $f(x)$ and $g(x)$ are two Weibull distributions defined as $$f(x;\lambda_1,k_1) =  \begin{cases}\frac{k_1}{\lambda_1}\left(\frac{x}{\lambda_1}\right)^{k_1-1}e^{-(x/\lambda_1)^{k_1}} & x\geq0 ,\\0 & x<0 ,\end{cases}$$ and $$f(x;\lambda_2,k_2) =  \begin{cases}\frac{k_2}{\lambda_2}\left(\frac{x}{\lambda_2}\right)^{k_2-1}e^{-(x/\lambda_2)^{k_2}} & x\geq0 ,\\0 & x<0 ,\end{cases}$$
is it possible to derive a metrics tensor in the space of probability as a function of the parameters $\lambda$ and $k$ using the Hellinger distance?
Thanks
 A: If we consider a new probability distribution defined through $f(x;\theta_i)$ and $g(x;\tilde\theta_i)$ given by $P(x;\theta_i,\tilde\theta_i)=\sqrt{f(x;\theta_i)g(x;\tilde\theta_i)}$, we can generalize the Fisher-Rao metric in the following way.
$$\frac{\partial H}{\partial\theta_i}=-\frac{1}{2}\int_\Omega dxP(x;\theta_i,\tilde\theta_i)\frac{\partial}{\partial\theta_i}\ln f(x;\theta_i)$$
$$\frac{\partial H}{\partial\tilde\theta_i}=-\frac{1}{2}\int_\Omega dxP(x;\theta_i,\tilde\theta_i)\frac{\partial}{\partial\tilde\theta_i}\ln g(x;\tilde\theta_i)$$
and so
$$\frac{\partial^2 H}{\partial\theta_i\partial\theta_j}=-\frac{1}{2}\int_\Omega dxP(x;\theta_i,\tilde\theta_i)\left[\frac{\partial^2}{\partial\theta_i\partial\theta_j}\ln f(x;\theta_i)+\frac{1}{4}\frac{\partial}{\partial\theta_i}\ln f(x;\theta_i)\frac{\partial}{\partial\theta_j}\ln f(x;\theta_i)\right]$$
and similarly
$$\frac{\partial^2 H}{\partial\tilde\theta_i\partial\tilde\theta_j}=-\frac{1}{2}\int_\Omega dxP(x;\theta_i,\tilde\theta_i)\left[\frac{\partial^2}{\partial\tilde\theta_i\partial\tilde\theta_j}\ln g(x;\tilde\theta_i)+\frac{1}{4}\frac{\partial}{\partial\tilde\theta_i}\ln g(x;\tilde\theta_i)\frac{\partial}{\partial\tilde\theta_j}\ln g(x;\tilde\theta_i)\right].$$
One has also cross products as
$$\frac{\partial^2 H}{\partial\theta_i\partial\tilde\theta_j}=-\frac{1}{4}\int_\Omega dxP(x;\theta_i,\tilde\theta_i)\frac{\partial}{\partial\theta_i}\ln f(x;\theta_i)\frac{\partial}{\partial\tilde\theta_j}\ln g(x;\tilde\theta_i).$$
Now, as done by Rao about Fisher information matrix, we can interpret these second derivatives of $H$ as the components of a metric tensor $h_{ij}(\theta,\tilde\theta)$ so that we can write down 
$$ds^2=h_{ij}(\theta,\tilde\theta)d\theta_id\tilde\theta_j$$
and work out Riemann geometry.
