Suppose that Θ , 𝑋1 , and 𝑋2 have zero means. Furthermore,

Question

Suppose that Θ , 𝑋1 , and 𝑋2 have zero means. Furthermore,

𝖵𝖺𝗋(𝑋1)=𝖵𝖺𝗋(𝑋2)=𝖵𝖺𝗋(Θ)=4,

and

𝖢𝗈𝗏(Θ,𝑋1)=𝖢𝗈𝗏(Θ,𝑋2)=𝖢𝗈𝗏(𝑋1,𝑋2)=1.

The LLMS estimator of Θ based on 𝑋1 and 𝑋2 is of the form Θˆ=𝑎1𝑋1+𝑎2𝑋2+𝑏 . Find the coefficients 𝑎1 , 𝑎2 , and 𝑏 . Hint: To find 𝑏 , recall the argument we used for the case of a single observation.

Answer 1

0.2

0.2

0

You are welcome!

Answer 2

To find the coefficients 𝑎1, 𝑎2, and 𝑏 for the LLMS estimator 𝚯̂ = 𝑎1𝑋1 + 𝑎2𝑋2 + 𝑏, we can use the following steps:

Step 1: Find the partial derivatives of the cost function with respect to 𝑎1, 𝑎2, and 𝑏.

Step 2: Set the partial derivatives equal to zero and solve the resulting equations.

Let's proceed with these steps:

Step 1: Find the partial derivatives of the cost function with respect to 𝑎1, 𝑎2, and 𝑏.

The cost function is given by:
𝐽(𝑎1, 𝑎2, 𝑏) = 𝐸[(Θ - 𝚯̂)^2]

Taking the partial derivative with respect to 𝑎1:
∂𝐽/∂𝑎1 = -2𝐸[𝑋1(Θ - 𝚯̂)]

Taking the partial derivative with respect to 𝑎2:
∂𝐽/∂𝑎2 = -2𝐸[𝑋2(Θ - 𝚯̂)]

Taking the partial derivative with respect to 𝑏:
∂𝐽/∂𝑏 = -2𝐸[Θ - 𝚯̂]

Step 2: Set the partial derivatives equal to zero and solve the resulting equations.

Setting ∂𝐽/∂𝑎1 = 0:
-2𝐸[𝑋1(Θ - 𝚯̂)] = 0
𝐸[𝑋1(Θ - 𝚯̂)] = 0

Since 𝐸[𝑋1(Θ - 𝚯̂)] = 𝐸[𝑋1(Θ - (𝑎1𝑋1 + 𝑎2𝑋2 + 𝑏))], we can expand the expression:
𝐸[Θ𝑋1 - 𝑎1𝑋1^2 - 𝑎2𝑋1𝑋2 - 𝑏𝑋1] = 0

Similarly, setting ∂𝐽/∂𝑎2 = 0 and ∂𝐽/∂𝑏 = 0, we get:
𝐸[Θ𝑋2 - 𝑎1𝑋1𝑋2 - 𝑎2𝑋2^2 - 𝑏𝑋2] = 0
𝐸[Θ - 𝑎1𝑋1 - 𝑎2𝑋2 - 𝑏] = 0

Simplifying these equations, we have:
𝐸[𝑋1(Θ - 𝑏)] = 𝐸[𝑋2(Θ - 𝑏)] = 𝐸[Θ - 𝑎1𝑋1 - 𝑎2𝑋2 - 𝑏] = 0

Since 𝑋1 and 𝑋2 have zero means, the equations become:
4(Θ - 𝑏) = 0
4(Θ - 𝑏) = 0
4 - 4𝑎1 - 4𝑎2 - 4𝑏 = 0

Simplifying further, we get:
Θ - 𝑏 = 0
Θ - 𝑏 = 0
1 - 𝑎1 - 𝑎2 - 𝑏 = 0

From the first two equations, we can conclude that Θ = 𝑏. Substituting this into the third equation, we have:
1 - 𝑎1 - 𝑎2 - Θ = 0

To solve this equation for 𝑎1 and 𝑎2, we can rearrange it as follows:
𝑎1 + 𝑎2 = 1 - Θ

Since 𝑎1 and 𝑎2 can be any real numbers, we can choose 𝑎1 = 1 - Θ and 𝑎2 = -Θ. This satisfies the equation and gives us the coefficients.

Finally, substituting Θ = 𝑏 into the LLMS estimator 𝚯̂ = 𝑎1𝑋1 + 𝑎2𝑋2 + 𝑏, we get:
𝚯̂ = (1 - Θ)𝑋1 - Θ𝑋2 + Θ

Therefore, the coefficients for the LLMS estimator are 𝑎1 = 1 - Θ, 𝑎2 = -Θ, and 𝑏 = Θ.

Answer 3

stop trying to disguise a homework dump by using different names.

As you may have guessed, the expertise level on statistics is a bit low here on this site. I had my one statistics class almost 50 years ago, so I'd have to study the subject again to answer these questions you post.

Clearly, it looks like you will have to do the same.