Risk function is defined as
$$R(\theta, \delta) := \int L(\theta, \delta(x)) P_{\theta}(dx)$$
where $x = (x_1, ... x_n)$ is an observation and $\delta(x)$ is a decision rule.
$P_{\theta}(x) = (\frac{1}{\sqrt{2\pi}})^n exp (\sum_{i=1}^n\frac{-(x_i-\theta)^2}{2}),$ $L(\theta, \delta) = (\theta - \delta)^2$ and $\delta(x) = \frac{1}{n}\sum_{i=1}^n{x_i}$.
- Is $\delta(x)$ a proper example of decision rule?
- Assuming that it is, how to calculate the value of $R$? I don't fully understant the notation and never seen an example, just theoretical definitions. (If answer to (1) is "no", can you provide an example of proper decision rule and then show how to calculate $R$?