You've assumed correctly. Sorry for the confusion, I am trying to solve this without giving too much information because I want to do it on my own but I need some guidance, I will reiterate: my pdf is f(x|theta) = e^-(x-theta). Problem asked to find the MLE and the MOM (method of moments) estimators for theta. Then, adjust those estimators so they are unbiased and find the relative efficiency of MLE to MOM. I've found MLE and MOM but I am having trouble understanding how I might go about adjusting X_(1) (which is my biased estimator of theta - theta_hat) so that it is unbiased. To restate your last reply, I need to find an unbiased estimator that is a function of my MLE, X_(1).