Moment estimator and its asymptotic distribution for exponential distribution

Question

I have this question: We let $X$ and $Y$ be independent exponentially distributed random variables with $E(X)=\beta$ and $E(Y)=\frac{1}{\beta}$,$\beta>1$. We let $(X_1,Y_1),\ldots,(X_n,Y_n)$ be a sample from this distribution.

1. How do I find the moment estimator $\hat{\beta}$ based on the statistic $t(x,y)=x-y$ and the sample $(X_1,Y_1),\ldots,(X_n,Y_n)$ and how do I find the moment estimator's asymptotic distribution?

I think the way to find the asymptotic distribution is to use that $V(t(x,y))/m'(\beta)^2$, but how do I find $m(\beta)$? And use that to find the moment estimator

Answer 1

I may be missing something? Let $a=\frac{\sum\limits_1^n t(X_k,Y_k)}{n}$ Then $\hat{\beta}$ can be estimated by $\hat{\beta}-\frac{1}{\hat{\beta}}=a$ or $\hat{\beta}=\frac{a+\sqrt{a^2+4}}{2}$.