I am trying to set up a hierarchical model for groups average estimation. I am confused when try to model the variance. Let me explain the problem:
Problem Definition:
Suppose we are interested on measure the average brake time of a set of cars. Let indexes the cars with $j$. For any car we have $n_j$ measures of the variable of interest, the brake time. Let be $X_{i,j}$ the brake time on the experiment $i$ for the car number $j$. Let's see an example:
$Car_1: 13, 45, 32, 12$
$Car_2: 32$
$Car_3: 13, 12, 11, 12, 16, 23$
The sample average break time of the $Car_2$ is 32 and I want to improve my parameter estimation with the information of the other cars. For such reason I want to set a hierarchical model.
Supposing that:
$X_{i,j} \sim N(\mu_{j}, \sigma_{j})$
I want to estimate $\mu_{j}$ and for do that I can assume that it comes from a overall distribution, in such way:
$\mu_{j} \sim N(\mu_0, \sigma_0)$
and for fitting the overall $\mu_0$ I will set up a weak prior distribution such that:
$\mu_0 \sim N(0,10000)$ # weak
Summarising, I am interested on setting up a hierarchical model for estimating the mean of each group. The problem arise when I try to deal with $\sigma_{j}$:
How to deal with the standar deviation $\sigma_{j}$? what non informative priors to use for $\sigma_{j}$? could I assume that all groups have the same standard deviation $\sigma_{j}$ = $\sigma$, and in such case, how to proceed?
In summary: how to model the $\sigma_{j}$ ?
Thanks