How to Compute the Mean of a Non-linear Estimator?

[mnb96]

Hi,
let's suppose we are given N statistically independent samples $x_1,\ldots,x_n$ from a certain distribution $f_X(x;\theta)$ depending on a parameter $\theta$.
We are also given an estimator for $\theta$ defined as follows:

$$\hat{\theta}}(N) = \min\{ x_i \\ : \\ i=1..N \}$$

How am I supposed to compute $$E\{ \hat{\theta}(N) \}$$?

I tried to apply the definition of mean-value as follows, but I can't go any further:

$$\int_{\mathbb{R}}\ldots\int_{\mathbb{R}} \min\{ x_1,\ldots,x_N \} \\ f_X(x_1)\ldots f_X(x_N)dx_1\ldots dx_N$$

Any idea?

 

[statdad]

Get the pdf of the minimum, and then compute the expected value with respect to that.

 

[mnb96]

Thanks.
I did as you suggested and succesfully solved the problem.