Suppose there is set of $k$ different countries. Let's say that the average income of a specific country can be given by:
$$X_i \sim N(\mu, \sigma_e^2 + \sigma_i^2)$$
Where:
- $X_i$ is the average income of a specific country
- $\mu$ is the average income of all countries within the set
- $\sigma_i^2$ is the variance of the average income for a specific country
- $\sigma_e^2$ describes how the mean of each country varies from the mean of all countries (i.e. relative to $\mu$)
Suppose I am interested in use Maximum Likelihood theory to estimate the values of $\mu$, $\sigma_i^2$, $\sigma_e^2$ . Note that in this question, you only have access to $X_i$ and $\sigma_i^2$ (i.e. you only have access to the mean and variance of the mean for each country). You do not have access to the individual earnings of citizens from these different countries - only the average incomes at the country level (and the variance of these measurements).
Since the average income of each country is Normally Distributed - I should be able to create a Probability Distribution corresponding to each country and multiply all of these distributions together. Then, I should be able to solve for the parameters of interest (i.e. $\mu$, $\sigma_i^2$, $\sigma_e^2$):
$$L(\theta) = \prod_{i=1}^k N(\mu, \sigma_e^2 + \sigma_i^2) = \prod_{i=1}^k \frac{1}{\sqrt{2\pi(\sigma_e^2 + \sigma_i^2)}} \exp{\left(-\frac{(x-\mu)^2}{2(\sigma_e^2 + \sigma_i^2)}\right)}$$
However, I do not think that the above logic is correct - this is because the probability distribution of each country follows a slightly different probability distribution.
As I understand, writing a Likelihood Function (i.e. joint product of Probability Distributions) requires each probability distribution being considered to be identical. Since in my problem this is not the case - I don't think that a valid Likelihood Function can be constructed and the parameters of interest $\theta = (\mu, \sigma_e^2, \sigma_1^2, \dots, \sigma_k^2)$ can not be estimated from this approach. This is because each probability function within the likelihood is essentially only based on a single observation.
Is my understanding correct?
Thanks!