I have the following problem: Let $X_1, \ldots, X_n$ be independent and identically distributed. The distribution of $X_i$ is a mixture of two Poisson distributions: $$ P\left[X_i=m\right]=\tau e^{-\lambda_1} \frac{\lambda_1^m}{m !}+(1-\tau) e^{-\lambda_2} \frac{\lambda_2^m}{m !}, \quad m \in \mathbb{N}_0, $$ where $\tau \in(0,1), \lambda_1>0, \lambda_2>0$ are unknown. Now i want to calculate the Moment estimators of the three unknown parameters.
Therefore I calculated the first three moments:
$\mathbb{E}[X]=\tau \lambda_1+ (1-\tau)\lambda_2$,
$\mathbb{E}[X^2]=\tau (\lambda_1+\lambda_1^2)+ (1-\tau)(\lambda_2+\lambda_2^2)$,
$\mathbb{E}[X^3]=\tau (\lambda_1+3\lambda_1^2+\lambda^3_1)+ (1-\tau)(\lambda_2+3\lambda_2^2+\lambda_2^3).$
However i didn't managed to solve those equations wrt. the parameters. Any suggestions how to start here? Or is there any better approach?