We let $X_1,...,X_n$ be i.i.d random variables from the pdf:
$f(x)=e^{-(x-\theta)}$ for $x\geq \theta$ and $f(x)=0$ otherwise.
Derive the method of moments estimator of $\theta$ and find its bias and variance. Is it a consistent estimator?
now ive worked out the expectation to be $-(\theta+1)$ and the variance to be $-2\theta^2-4\theta-3$
From the formula ive seen for the method of moments estimator ive tried to work it out and i got it to be $1-\frac{1}{n}\sum_{k=1}^{n}(x_i)$ but i am not sure, and im not quite sure how to work out the bias and variance or check for consistency.