The distribution when combining two samples together?

Question

Suppose $X\sim N(0,{\sigma}^2)$ and $Y\sim N(0,{2\sigma}^2)$ . $X_1, ..., X_m$ are the samples from $X$ and $Y_1, ..., Y_n$ are the samples from $Y$. And then combine two samples as a new sample $\{X_1, ..., X_m,Y_1, ..., Y_n\}$. The question is what's the distribution of the new sample? And how to calculate the UMVUE of $\sigma^2$? How to calculate the Fisher Information of $\sigma^2$?

Answer 1

Set $$X_{m+1}=\frac{Y_1}{2},...,X_{m+n}=\frac{Y_n}{2}$$ Now you have an i.i.d. sample of size $(m+n)$ from $N(0,\sigma^2)$. The theory in that standard case follows with no modification. An estimator is going to be a function depending on the sample $X_1,...,X_m,Y_1,...,Y_n$ (which is exactly the same as $X_1,...,X_{m+n}$ because you applied a deterministic transformation meaning that if you know the first, then you can get the second with no extra "information"). Now, think about the variane estimator $$ \frac{1}{m + n - 1} \left( \sum_{i = 1}^{m + n} (X_i^{} - \bar{X})^2 \right) $$ where $$ \bar{X} = \frac{1}{m + n} \sum_{i = 1}^{m + n} X_i $$