Suppose that instead of estimators of the form aX+e , we consider estimators of the form Θˆ=aX and ask for the value of a that minimizes the mean squared error. Mimic the derivation you have just seen and find the optimal value of a . Your answer should be an algebraic expression involving some of the constants b , c , d , where b=E[Θ2] , c=E[ΘX] , d=E[X2] .

Question

Suppose that instead of estimators of the form aX+e , we consider estimators of the form Θˆ=aX and ask for the value of a that minimizes the mean squared error. Mimic the derivation you have just seen and find the optimal value of a . Your answer should be an algebraic expression involving some of the constants b , c , d , where b=E[Θ2] , c=E[ΘX] , d=E[X2] .

Answer 1

Well, let's dive into algebraic fun land and find that optimal value of a.

Starting with the mean squared error (MSE), we have:

MSE = E[(Θˆ - Θ)^2]
= E[((aX) - Θ)^2]
= E[(aX - Θ)^2]
= E[a^2X^2 - 2aXΘ + Θ^2]
= a^2E[X^2] - 2aE[XΘ] + E[Θ^2]

Now, we want to find the value of a that minimizes the MSE. To do that, we take the derivative of the MSE with respect to a and set it equal to zero:

d(MSE)/da = 2aE[X^2] - 2E[XΘ] = 0

Dividing through by 2, we get:

aE[X^2] - E[XΘ] = 0

Now, we want to solve for a, so let's isolate it:

aE[X^2] = E[XΘ]
a = E[XΘ] / E[X^2]

There you have it! The optimal value of a is given by:

a = E[XΘ] / E[X^2]

This expression involves the constants b, c, and d, which are defined as:

b = E[Θ^2]
c = E[ΘX]
d = E[X^2]

So, the optimal value of a is:

a = c / d

Hope that puts a smile on your face while dealing with estimators!

Answer 2

To find the optimal value of a that minimizes the mean squared error (MSE) when using estimators of the form Θˆ = aX, we need to minimize the expression for MSE.

The mean squared error (MSE) is defined as:
MSE = E[(Θˆ - Θ)^2]

Substituting Θˆ = aX, we have:
MSE = E[(aX - Θ)^2]

Expanding the square and using linearity of expectation, we get:
MSE = E[a^2X^2 - 2aXΘ + Θ^2]
= a^2E[X^2] - 2aE[XΘ] + E[Θ^2] (taking expectations of each term)

Now we need to minimize the MSE with respect to a. Taking the derivative of the MSE with respect to a and setting it to zero, we have:
d(MSE)/da = 2aE[X^2] - 2E[XΘ] = 0

Simplifying the equation, we have:
aE[X^2] = E[XΘ]

Finally, solving for a, we get:
a = E[XΘ] / E[X^2]

Therefore, the optimal value of a that minimizes the mean squared error (MSE) is given by:
a = E[XΘ] / E[X^2]

Note: The expression involves the constants b = E[Θ^2], c = E[ΘX], and d = E[X^2].

Answer 3

To find the optimal value of a that minimizes the mean squared error, we need to minimize the expression for the mean squared error (MSE) with respect to a. The MSE is given by:

MSE = E[(Θˆ - Θ)²]

Substituting Θˆ = aX into the above equation:

MSE = E[(aX - Θ)²]

Expanding the square and rearranging terms:

MSE = E[a²X² - 2aXΘ + Θ²]

Now, let's consider each term separately:

1. E[a²X²] = a²E[X²] = ad (using the given information that E[X²] = d)

2. E[-2aXΘ] = -2aE[XΘ] = -2ac (using the given information that E[ΘX] = c)

3. E[Θ²] = b (using the given information that E[Θ²] = b)

Putting it all together, we have:

MSE = ad - 2ac + b

To find the optimal value of a, we differentiate the MSE with respect to a and set it to zero. Differentiating the MSE with respect to a:

d/d(a) (MSE) = 2ad - 2c = 0

Solving for a:

2ad = 2c

a = c/d

Therefore, the optimal value of a that minimizes the mean squared error is a = c/d. Note that this expression involves the constants c and d, where c = E[ΘX] and d = E[X²].