Question: A 95 -year old man with a tumor in his lung must decide between two options of (1) receiving radiotherapy or (2) no treatment. He would like to maximize his quality adjusted life expectancy which is defined as the expected length of time he survives minus 3 months if he goes through radiotherapy. If the tumor is benign, the man's life expectancy is 35 months. If the tumor is malignant and he receives radiotherapy, his life expectancy is 16 months. If the tumor is malignant, and he receives no treatment, his life expectancy is 6 months. A test called a bronchoscopy is performed to test if the tumor is malignant. The test has a 70% chance of detecting a malignant tumor if the tumor is truly malignant and a 2% chance of falsely claiming a malignant tumor if the tumor is truly benign.

1. Show that the man’s loss function is
   
2. Find the Bayes decision procedure if the man‘s prior is that he has a 90% chance of having a malignant tumor (the decision procedure should specify what the man will do if the test claims a malignant tumor and what the man will do if the test claims a benign tumor).
3. Among the nonrandomized decision procedures, find the minimax decision procedure.
4. The risk set \(S\) is the set of possible risk functions of all decision procedures, \(S=\left\{\left(R\left(\theta_{1}, \delta\right), R\left(\theta_{2}, \delta\right)\right): \delta \in D^{*}\right\}\), where \(D^{*}\) is the set of all decision procedures, including randomized ones. Make a plot of the risk set.
5. Find the minimax decision procedure over all decision procedures, including randomized ones. Justify your answer.

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