Consider the following set of $N$ Independent Random Variables:
- $X_1 = \{x_{11}, x_{12}, \dots x_{1n_1}\}$
- $X_2 = \{x_{21}, x_{22}, \dots, x_{2n_2}\}$
- etc.
- $X_N = \{x_{N1}, x_{N2}, \dots, x_{Nn_n}\}$
Each of these Random Variables has:
- A sample size of $N_i$ random realizations (i.e. $X_1$ has $n_1$ realizations, $X_2$ has $n_2$ realizations, etc.)
- Unknown Probability Distributions
Using the Central Limit Theorem, we can argue that regardless of the underlying Probability Distribution:
$$E(X_i) = \sum_{i=1}^{n_i} \frac{x_i}{n_i} = \mu_i$$ $$E(X_i^2) - E(X_i)^2 = \sum_{i=1}^{n} \frac{(x_i - \mu_i)^2}{n} = \sigma^2_i $$ $$\mu_i \sim N(\mu_i, \sigma^2_i)$$
By extension, I could then say that if I define a new Random Variable $W$:
$$W = \frac{\mu_1 + \mu_2 + \dots + \mu_N}{N}$$
$$ E(W) = \frac{1}{N} \left( E(\mu_1) + E(\mu_2) + \dots + E(\mu_N) \right) = \frac{1}{N} \left( \mu_1 + \mu_2 + \dots + \mu_N \right) = \sum_{i=1}^{N} \frac{\mu_i}{N}$$
$$ Var(W) = \frac{1}{N^2} \left( Var(\mu_1) + Var(\mu_2) + \dots + Var(\mu_N) \right) = \frac{1}{N^2} \left( \sigma^2_1 + \sigma^2_2 + \dots + \sigma^2_N \right) = \frac{1}{N^2} \sum_{i=1}^{N} \sigma^2_i $$
And then again using Central Limit Theorem, we could again argue that: $$W \sim N \left( \sum_{i=1}^{N} \frac{\mu_i}{N}, \frac{1}{N^2} \sum_{i=1}^{N} \sigma^2_i \right)$$
My Question: Given that each of these random variables $X_i$ have a large enough sample size, can $W$ be considered as a "suitable" (i.e. dominant estimator, admissible, statistical properties such as unbiasedness, consistency, minimum variance, etc.) estimator for estimating the population mean instead of using the Weighted Mean (e.g. weighting proportional to counts $n_i$, weighting proportional to $\sigma^2_i$)?
Thanks!