Theory and Methods of Statistics

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 H₀: θ ≤ θ₀ and H₁: θ > θ₀ (for a given θ₀) 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 H₀ if and only if ${\text{E}}_G\left[\theta |X=x\right]>$