question archive A statistician has to decide on the basis of one observation whether the parameter ???? of a Bernoulli distribution is 1/4 or 1/2 ; his loss ( a penalty that is deducted from his fee ) is $80 if he is wrong
A statistician has to decide on the basis of one observation whether the parameter ???? of a Bernoulli distribution is 1/4 or 1/2 ; his loss ( a penalty that is deducted from his fee ) is $80 if he is wrong
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A statistician has to decide on the basis of one observation whether the parameter ???? of a Bernoulli distribution is 1/4 or 1/2 ;
his loss ( a penalty that is deducted from his fee ) is $80 if he is wrong.
(a) Construct a table showing the four possible values of the loss function.
(b) List the four possible decision functions.
(c) Construct a table showing all the values of the corresponding risk function.
(d) Find the decision function that is the best according to the minimax criterion.
(e) Find the decision function that is the best according to the Bayes criterion if the probabilities assigned to ???? = 1/4 and ???? = 1/2 are, respectively, 2/3 and 1/3.
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Please see the given below explanation and answer and let me know if you have any doubt.
Step-by-step explanation
According to the question,
Here statistican has to decide the parameter of binomial
There are two outcomes namely 1/2 or 1/4
Hence both are equally likely with p = 1/2 and q = 1/2
n =2
(a) Construct a table showing the four possible values of the loss function.
The four possible values are:
| profit | loss | |
| p1 | 1/4 | 3/4 |
| p2 | 1/2 | 1/2 |
(b) List the four possible decision functions.
1.accept when p=1/4 , reject when p=3/4
2.accept when p=1/2, reject when p=1/2
3.accept when p=3/4 ,reject when p=3/4
4.accept when p=1/4,reject when p=1/4
(C)
|
1.accept when p=1/4 , reject when p=3/4
2.accept when p=1/2, reject when p=1/2
3.accept when p=3/4 ,reject when p=3/4
4.accept when p=1/4,reject when p=1/4
|
(d) Find the decision function that is the best according to the minimax criterion.
The trials n must be fixed to 50 or more .the probability with p=1/2 can be the best decision.
