Minimize loss function

Question

Given a random sample $X_1, \dots, X_n$, find the estimator of position $\theta$ that minimize the following loss function

$$g(\theta) = \sum_{i=1}^n (X_i - \theta)^2 ,\quad \quad g(\theta)=\sum_{i=1}^n |X_i - \theta|$$

Now, I've already now what a loss function is, and the definition of risk and minimax. And I just read a couple of examples which gives you a random sample with a fixed distribution and it compares two distint estimators using the risk. But I can't understand the idea of this exercise (and I don't have a rough definition of what "estimator of position" is).

I don't want the answer, just want to know what am I have to do. Any tips?

Thank you

Answer 1

Estimation of position (or a more common term, estimator for $\theta$, usually denoted as $\hat{\theta}$) means

$$\hat{\theta} = \arg \min_{\theta} g(\theta).$$

I think the purpose of the exercise is to build your intuition on the choice of loss function.