Exercises

4.1. Suppose that X is distributed with a pdf fx,θsi205_e where θ is an unknown real parameter. Consider the two-decision problem of choosing between the hypotheses H0: θθ0 and H1: θ > θ0 (for a given θ0) with the loss function

Lθ,a0=θθ0+,Lθ,a1=θ0θ+,

si328_e

where for any real number x, x+ denotes maxx,0si329_e, and ai is the action to accept Hi, i = 0,1. Show that the Bayes rule with respect to a prior cdf G of θ rejects H0 if and only if EGθ|X=x>

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