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Let $X_1,\ldots,X_n$ be i.i.d. $\mathsf{Exp}(\lambda)$ random variables, where $\lambda$ is unknown. Consider $f_{\min}(x) = \min_{i}(X_i)=$ $ n \lambda $ Exp$(n\lambda x)$.

I am told that $\hat \theta \mathrel := n \cdot \min_i(X_i)$ is an unbiased estimator for the parameter $1/\lambda$. Indeed, this is true since the expected value of the above defined $\hat \theta$ is equal to $1/\lambda$. But, in this setting how would one proceed to compute $1/\lambda$ from $\hat \theta$? (don't we construct an estimator in order to compute a parameter?)

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...in this setting how would one proceed to compute $1/\lambda$ from $\hat \theta$?

You can't compute $1/\lambda$ because $\lambda$ is the unknown parameter. That is why we estimate it. We can take the value $\hat{\theta}(\mathbf{x}) = n x_{(1)}$ to be an estimate of $1/\lambda$ (and we can compute that estimate from the observed data), but we will not know the true value of the parameter.

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  • $\begingroup$ thanks for commenting. Indeed, that was I meant, e.g. how to compute an estimate of it. I thnk I got confused thinking about cases where we use the sample mean as an estimator. I see it more clearly now. By the way, $\hat \theta$ has a quadratic risk of $1/\lambda^2$. Since it's unbiased is it also consistent? $\endgroup$ – GunnRos86 Mar 4 '19 at 12:25
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You know that $\hat\theta(X_1,\ldots,X_n)=n\min_\limits{1\le i\le n} X_i$ is an unbiased estimator of $\frac{1}{\lambda}$, a function of the unknown parameter $\lambda$.

This implies that an unbiased estimate of $\frac{1}{\lambda}$ is given by $\hat\theta(x_1,\ldots,x_n)=n\min_\limits{1\le i\le n} x_i$. [Here $x_i$ denotes the observed value of $X_i$ in the sample]. This is the value of $\frac{1}{\lambda}$ you are seeking based on the observed sample $x_1,\ldots,x_n$.

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    $\begingroup$ I'm not sure I understand the distinction you're trying to make here. The concept of "bias" applies only to an estimator, not an estimate. Can you clarify? $\endgroup$ – Ben Mar 4 '19 at 11:01
  • $\begingroup$ I am not making any distinction; it is just a way of saying that $\frac{1}{\lambda}$ is unbiasedly estimated by $\hat\theta$. $\endgroup$ – StubbornAtom Mar 4 '19 at 11:21

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