A statistician has to decide on the basis of one obser-vation whether the parameter θ of a Bernoulli distribu-tion is 0, 1 2 , or 1; her loss in dollars (a penalty that isdeducted from her fee) is 100 times the absolute valueof her error.(a) Construct a table showing the nine possible values ofthe loss function. (b) List the nine possible decision functions and con-struct a table showing all the values of the corresponding risk function. (c) Show that five of the decision functions are not admis-sible and that, according to the minimax criterion, the remaining decision functions are all equally good.(d) Which decision function is best, according to the Bayes criterion, if the three possible values of the param-eter θ are regarded as equally likely?
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
A statistician has to decide on the basis of one obser-
vation whether the parameter θ of a Bernoulli distribu-
tion is 0, 1
2 , or 1; her loss in dollars (a penalty that is
deducted from her fee) is 100 times the absolute value
of her error.
(a) Construct a table showing the nine possible values of
the loss function.
(b) List the nine possible decision
struct a table showing all the values of the corresponding
risk function.
(c) Show that five of the decision functions are not admis-
sible and that, according to the minimax criterion, the
remaining decision functions are all equally good.
(d) Which decision function is best, according to the
Bayes criterion, if the three possible values of the param-
eter θ are regarded as equally likely?
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