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

#### revans.evans

##### New Member
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).

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

##### New Member
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 $$X_{(1)}$$).