Consider a random sample of size of from a distribution with its pdf given by
()={
(1/)^(−(−)/ ), >
0, ℎ
Find the MMEs of and .
I found the MME of as barX(n) - using integration to find E(X), which is +. Can I use this to find that of , which is barX(n)-?
Btw do you think I am on the correct approach?
We can calculate $E(X)=\int_{\alpha}^{\infty} x\frac{1}{\theta}e^{-\frac{x-\alpha}{\theta}}dx=\alpha+\theta$ and $E(X^2)=\int_{\alpha}^{\infty} x^2\frac{1}{\theta}e^{-\frac{x-\alpha}{\theta}}dx=\alpha^2+\theta^2+2\alpha\theta$ and $Var(X)=E(X^2)-E^2(X)=\theta^2$
So we have $\tilde{\theta}=\pm\sqrt{\overline{X^2}-\overline{X}^2}$ since $\theta > 0$ Therefore MME of $\theta$ is $\tilde{\theta}=\sqrt{\overline{X^2}-\overline{X}^2}$and for MME of $\alpha$, we can find above equations $\tilde{\alpha}=\overline{X}-\tilde{\theta}=\overline{X}-\sqrt{\overline{X^2}-\overline{X}^2}$