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I have the following density:

$f(x)=\frac{1}{\sigma}e^{\frac{-(x-\theta)}{\sigma}}$ where $x>\theta$. It is required to obtain the maximum likelihood estimator and method of moment estimator for $\sigma$ and $\theta$. I have obtained the method of moment estimator by calculating the first and second moment as follows:

$E(X)=\theta+\sigma$, $E(X^2)=2\sigma^2+\theta^2+2\theta\sigma$

After a little simplification, the estimates turned out to be:

$\sigma=\sqrt{E(X^2)-(E(X))^2}$ and $\theta=E(X)-\sigma$.

I am not sure whether my estimates are correct. I have obtained the maximum likelihood estimator and they matched with what textbook calculated but method of moment estimator is something different according to textbook. Please take a moment to check where I was wrong. Thanks in advance.

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  • $\begingroup$ Where are your sample of X values? Both the MLE and MM are based on the sample values. $\endgroup$ – Michael Chernick Jan 28 '18 at 8:55
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The easiest thing here is to note that $Y \equiv X - \theta \sim \text{Exp}(\sigma)$. So you can use the standard moments of the exponential distribution, then adjust them to account for adding $\theta$. You have:

$$\mathbb{E}(X) = \theta + \mathbb{E}(Y) = \theta + \sigma,$$

$$\mathbb{V}(X) = \mathbb{E}(Y) = \sigma^2.$$

These are the actual moments, so the method-of-moment (MOM) estimators follow by substitution of actual moments with sample moments.

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