Reference prior of normal distribution with unknow mean and variance

Problem:

Assume that $$X|\theta \sim N(\theta, \sigma^2)$$ for unknow $$\theta$$, and unknown $$\sigma$$.

a. Find the reference priors of $$(\theta, \sigma)$$, when $$\sigma$$ is of interest.

b. Find the reference priors of $$(\theta, \sigma^2)$$, when $$\sigma^2$$ is of interest.

My attempt:

We have the fisher information,

$$I(\theta, \sigma^2) =-E\begin{bmatrix} \frac{\partial^2 l}{\partial \theta^2} &\frac{\partial^2 l}{\partial \theta \partial \sigma} \\ \frac{\partial^2 l}{\partial \sigma \partial \theta} & \frac{\partial^2 l}{\partial \sigma^2} \end{bmatrix}\\ =\begin{bmatrix} \frac{-1}{\sigma^2} &\frac{-2(a-\theta)}{\sigma^3} \\ \frac{-2(a-\theta)}{\sigma^3} & \frac{-3(a-\theta)}{\sigma^4}+\frac{1}{\sigma^2} \end{bmatrix}\\ =\begin{bmatrix} \frac{1}{\sigma^2} &0 \\ 0 & \frac{2}{\sigma^2} \end{bmatrix}$$

Now when $$\theta_1=\sigma$$,

$$\pi_R(\theta, \sigma) \propto \frac{1}{\sigma}$$

Now when $$\theta_1=\sigma^2$$,

$$\pi_R(\theta, \sigma^2) \propto \pi_R(\theta, \sigma) |\frac{\partial (\theta, \sigma)}{\partial (\theta, \sigma^2)} |$$

I can't make sense of $$|\frac{\partial (\theta, \sigma)}{\partial (\theta, \sigma^2)}|$$. How can I compute this expression? Also, not sure how reference prior is different than Jeffrey's prior! I appreciate your suggestions. Thanks!

• Is the reference prior the same as Jeffreys prior ? Oct 18 '21 at 17:02
• @Pohoua I had the same question! I am not sure! Oct 18 '21 at 18:19
• No, not at all!, the reference prior is the Jeffreys prior on the parameter of interest when the nuisance parameter is integrated out using the Jeffreys prior on this nuisance parameter conditional on the parameter of interest!!! Oct 18 '21 at 19:57
• The expression $|\partial(\theta,\sigma)/\partial(\theta,\sigma^2)|$ seems to be the Jacobian of the transformation $(\theta,\sigma)\rightarrow(\theta,\sigma^2)$. Oct 19 '21 at 3:53
• I think $1/sigma$ is the Jefferys prior on $(\theta,\sigma)$. Reference prior and Jefferys prior should be equal in your case since you do not have any nuisance parameters and you mle would satisfy the the conditions of asymptotic Normality. But I am not sure about the last bit. Oct 19 '21 at 4:00