obtaining an unbiased estimator for the min: x_(1)

[revans.evans]

How would I go about finding an unbiased estimator for the minimum order statistic for a given PDF and distribution.

Example:

f(x|theta)=e^-(x-theta) for x > theta

found my MLE to be X_(1), the MIN(X_i). Setting my theta_hat = X_(1) and plugging in to n*f(x)*[1-F(x)]^(n-1) I obtain n*e^n(theta-x).

I am not sure if I am going about this the correct way though.

Thank you and let me know if there is any more information to provide.

 

[Dason]

Your question as worded doesn't make much sense. The minimum order statistic is a statistic and as such... you don't need to estimate it. This would be like asking how to find an unbiased estimator for \(\bar{x}\).

 

[revans.evans]

Well, that makes sense. How about this: so I've found X_(1) to be the statistic for which the distribution function is maximized. So, I set theta_hat equal to X_(1) and that is my MLE. I am then asked to adjust this so it is an unbiased estimator. My solution was to find X_(1) = n*e^n(theta-x) and try to adjust this. Does this make more sense?

 

[Dason]

I'm assuming you're talking about the likelihood function (not the distribution function) and I'm assuming you're trying to estimate \(\theta\).

Is your task just to find an unbiased estimator or do you need to find an unbiased estimator that is a function of your MLE (in this case \(X_{(1)}\)).

 

[revans.evans]

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).