I am trying to do a statistical hypothesis testing between two random variables $Y^{(1)}$ and $Y^{(2)}$:
$Y^{(1)} = \frac{\sum_{i=1}^{n}w_i^{(1)}X_i^{(1)}}{\sum_{i=1}^{n}X_i^{(1)}}$
$Y^{(2)} = \frac{\sum_{i=1}^{n}w_i^{(2)}X_i^{(2)}}{\sum_{i=1}^{n}X_i^{(2)}}$
both of which are weighted average of a series of independent random variables $X_i$. Each $X_i$ is a negative binomial distribution with different but known parameters. All the weights $w_i$ are known.
Does anyone know what the distribution of $Y$ is?
Any ideas for the hypothesis testing?