I have $f_{\tau, \theta}(y)=\theta e^{-\theta(y-\tau)}, y\ge\tau, \theta\gt 0$. Assume both parameters unknown.
$\mu_1=E(Y)=\tau+\frac1\theta=\bar{Y}=m_1$ where $m$ is the sample moment.
$\mu_2=E(Y^2)=(E(Y))^2+Var(Y)=(\tau+\frac1\theta)^2+\frac{1}{\theta^2}=\frac1n \sum Y_i^2=m_2$.
$\mu_2-\mu_1^2=Var(Y)=\frac{1}{\theta^2}=(\frac1n \sum Y_i^2)-{\bar{Y}}^2=\frac1n\sum(Y_i-\bar{Y})^2\implies \hat{\theta}=\sqrt{\frac{n}{\sum(Y_i-\bar{Y})^2}}$
Then substitute this result into $\mu_1$, we have $\hat\tau=\bar Y-\sqrt{\frac{\sum(Y_i-\bar{Y})^2}{n}}$
Is my procedure correct?
Best Answer
There is a small problem in your notation, as $\mu_1 =\overline Y$ does not hold. The term on the right-hand side is simply the estimator for $\mu_1$ (and similarily later). But your estimators are correct for $\tau, \theta$ are correct.