Hi, I need help for Take Home Midterm, Statistics 550, Fall 2006, Question #3

.A possibly biased coin has an unknown probability of landing heads, where is known to equal either .The coin is tossed once, and you observe the outcome.You then have a choice whether or not to play a certain game.In the game, you predict the next toss.If you predict correctly, you are paid $2; if you predict incorrectly, you lose $1.If you don't play the game, you lose nothing and gain nothing.You thus have three possible actions -- : play and predict heads; : play and predict tails; don't play.There are nine non_randomized decision rules shown in the following table that prescribe what action to take based on whether the first toss is heads or tails:

First Toss Heads Tails

d1 a1 a1

d2 a1 a2

d3 a1 a3

d4 a2 a1

d5 a2 a2

d6 a2 a3

d7 a3 a1

d8 a3 a2

d9 a3 a3

(a) What are the risk functions for the nine decision rules?

(b) What is (are) the minimax rule(s)?

(c) For the following prior distribution on , what is (are) the Bayes rule(s)?

(d) Which rules are admissible and which rules are inadmissible?