## Exercises

4.1. Suppose that X is distributed with a pdf $f\left(x,\theta \right)$ where θ is an unknown real parameter. Consider the two-decision problem of choosing between the hypotheses H_{0}: θ ≤ θ_{0} and H_{1}: θ > θ_{0} (for a given θ_{0}) with the loss function

$\begin{array}{l}\hfill L\left(\theta ,{a}_{0}\right)={\left(\theta -{\theta}_{0}\right)}_{+},L\left(\theta ,{a}_{1}\right)={\left({\theta}_{0}-\theta \right)}_{+},\end{array}$

where for any real number x, x_{+} denotes $max\left(x,0\right)$, and a_{i} is the action to accept H_{i}, i = 0,1. Show that the Bayes rule with respect to a prior cdf G of θ rejects H_{0} if and only if ${\text{E}}_{G}\left[\theta |X=x\right]>$

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