A statistician has to decide on the basis of one observation whether the parameter θ of a Bernoulli distribution is 0, 1/2, or 1; her loss in dollars (a penalty that is deducted from her fee) is 100 times the absolute value of her error. (a) Construct a table showing the nine possible values of the loss function. (b) List the nine possible decision functions and construct a table showing all the values of the corresponding risk function. (c) Show that five of the decision functions are not admissible and that, according to the mini max criterion, the remaining decision functions are all equally good. (d) Which decision function is best, according to the Bayes criterion, if the three possible values of the parameter θ are regarded as equally likely?
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