Max Randomised Action Risk Function Worksheet

A pharmaceutical company produces a new painkiller in batches of 10,000. Due to the
large sample space and inefficiency, 100 samples are taken and tested each time. The
test result may be of three types:
?1 is nsufficient concentration¿2 is ligible concentration¿3 is xcellent concentration”o avoid misunderstandings, all drugs that reach the calibrated concentrations do not
pose a threat to the patient’s health, but higher concentrations represent a higher cost
for the reagents invested.
The company has produced a particular batch and is now deciding what action to take:
a1 3end to market, or a2 – throw away.
The losses corresponding to each action ai , i = 1,2, and types ?j , j= 1, 2, 3, are
represented by the following loss matrix:
Table 1: Loss matrix
a1 a2
?1
4
0
?2
1
1
?3
0
5
By definition the loss function ! (“, #? (%, ?)) is: ! (“, #? (%, ?)) = &#?(%, ?)!(“, ‘)
And the risk function ( (“, #?) of a randomized decision rule #? (%, ?) with the loss
function
! (“, #? (%, ?)) is: ( (“, #?) = &!” [!(“, # ? (,)]
By assuming the #? (a1) =/, and #? (a2) =1?/, we can obtain the Risk function
( (“, #?) = &!” [!(“, # ? (,)]
= #? (a1) ! (“, a1) + #?(a2) ! (“, a2)
= /?! (“, a1) + (1?/) ?! (“, a2)
4/ + 0 ? (1 ? /) = 4/ 89 ” = “$
89 ” = “%
=0 / + (1 ? /) = 1
0 + 5(1 ? /) = 5 ? 5/
89 ” = “&
~
‘
‘
If p ? ( , then 4p ? 5-5p; If p ? ( , then 4p ? 5-5p;
‘
So when p = ( , we obtain the minimum of maximum loss
Letàpoint it by graph p ? [0,1]
X
i
1
maximum
~
–
0 02
=
0 83
) probability
=
0.2
0.4
0.60.8
loss
1
Then, itàclear that Sup) (B”, #** C = D’%4/, 1,5 ? 4/.
‘
*
Thus, min{Sup) (B”, #** C}=Sup) (B”, #’/(
C= ( *4 =
%(
*
In conclusion, the minimax randomized action is #’/(
,the minimax value is
%(
.

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