Risk Theory-computing the risk function

[act3]

Homework Statement

Let X~U(0,1). Let the loss function be L(x,y) = (x-y)^4.
a. Compute the risk function R(y) here explicitly. What kind of function is it here?

Homework Equations

let Y=1/2

The Attempt at a Solution

I know you have to find the first and second derivatives of the loss function.
dL(X,y)/dy= -4(x-y)^3
d^2L(X,y)/dy^2= 12(x-y)^2

I don't know where to go from here though. Any help would be much appreciated!

 

[mighty2000]

The risk function, denoted by R(y), is a function that measures the expected loss when a particular value of y is chosen as the estimate for the true value of X. In this case, since X is uniformly distributed between 0 and 1, we can express the risk function as follows:

R(y) = E[L(X,y)] = ∫₀¹ (x-y)^4 dx

To evaluate this integral, we can use the fact that X is uniformly distributed between 0 and 1, so its probability density function is given by f(x) = 1 for 0 ≤ x ≤ 1. Therefore, the expected value of X is given by:

E[X] = ∫₀¹ x f(x) dx = ∫₀¹ x dx = 1/2

Now, we can use this result to evaluate the risk function:

R(y) = ∫₀¹ (x-y)^4 dx = ∫₀¹ (x^4 - 4x^3y + 6x^2y^2 - 4xy^3 + y^4) dx

= 1/5 - 4/4(1/2)y + 6/3(1/2)^2y^2 - 4/2(1/2)^3y^3 + 1/1(1/2)^4y^4

= 1/5 - y + 3/8y^2 - 1/4y^3 + 1/16y^4

Therefore, the risk function is a polynomial function of degree 4.