# Question: A statistician has to decide on the basis of two

A statistician has to decide on the basis of two observations whether the parameter θ of a binomial distribution is 1/4 or 1/2 ; his loss (a penalty that is deducted from his fee) is $ 160 if he is wrong.

(a) Construct a table showing the four possible values of the loss function.

(b) List the eight possible decision functions and construct a table showing all the values of the corresponding risk function.

(c) Show that three of the decision functions are not admissible.

(d) Find the decision function that is best according to the minimax criterion.

(e) Find the decision function that is best according to the Bayes criterion if the probabilities assigned to θ = 1/4 and θ = 1/2 are, respectively, 2/3 and 1/3.

(a) Construct a table showing the four possible values of the loss function.

(b) List the eight possible decision functions and construct a table showing all the values of the corresponding risk function.

(c) Show that three of the decision functions are not admissible.

(d) Find the decision function that is best according to the minimax criterion.

(e) Find the decision function that is best according to the Bayes criterion if the probabilities assigned to θ = 1/4 and θ = 1/2 are, respectively, 2/3 and 1/3.

## Answer to relevant Questions

With reference to the illustration on page 267, show that even if the coin is flipped n times, there are only two admissible decision functions. Also, construct a table showing the values of the risk function corresponding ...A statistician has to decide on the basis of a single observation whether the parameter θ of the density Equals θ1 or θ2, where θ1 < θ2. If he decides on θ1 when the observed value is less than the constant k, on θ2 ...If X is a random variable having the binomial distribution with the parameters n and θ, show that n ∙ X/n ∙ (1 – X/n) is a biased estimator of the variance of X. Rework Example 10.5 using the alternative formula for the information given in Exercise 10.19. Example 10.5 Show that is a minimum variance unbiased estimator of the mean µ of a normal population. With reference to Exercise 10.12, show that 2X – 1 is also an unbiased estimator of k, and find the efficiency of this estimator relative to the one of part (b) of Exercise 10.12 for (a) n = 2; (b) n = 3.Post your question