To find (I(θ))11, we need to determine the first diagonal entry of the Fisher information matrix.
The Fisher information matrix for a two-dimensional parameter θ = (μ, σ^2)^T is given by:
I(θ) = [[I11, I12],
[I21, I22]]
where Iij is the (i,j)th entry of the matrix.
In this case, we have:
I(θ) = [[n/σ^2, 0],
[0, n/(2σ^4)]]
So, (I(θ))11 = n/σ^2.
To find (I(θ))22, we need to determine the second diagonal entry of the Fisher information matrix.
In this case, (I(θ))22 = n/(2σ^4).
Now, to determine Jeffrey's prior, we need to calculate the square root of the determinant of the Fisher information matrix:
√(det(I(θ))) = √(n/σ^2 * n/(2σ^4)) = √(n^2/(2σ^6)) = n/(√(2)σ^3)
Jeffrey's prior, π(θ), is then given by:
π(θ) = k * √(det(I(θ)))
where k is a constant.
To satisfy π((1, 1)^T) = 1, we substitute μ = 1 and σ = 1 into the expression for π(θ):
π((1, 1)^T) = k * √(1/(√(2))) = k/(√(2)) = 1
Solving for k, we find:
k = √(2)
Therefore, Jeffrey's prior, π(θ), in terms of μ and σ, is:
π(θ) = √(2) * √(det(I(θ))) = √(2) * (n/(√(2)σ^3)) = n/σ^3.
In this problem we will consider a model which has a two-dimensional parameter. Then you will calculate Jeffrey's prior using the Fisher information matrix.
Suppose that X_1, \ldots , X_ n \stackrel{iid}{\sim } N(\mu ,\sigma ^2), where \mu and \sigma ^2 are unknown. In this case, the Fisher information matrix I(\theta ) for \theta =(\mu ,\sigma ^2)^ T will be a 2\times 2 matrix, where the off-diagonal entries are 0.
Find (I(\theta ))_{11}.
n/theta^2
unanswered
Find (I(\theta ))_{22}.
n/(2*theta^4)
unanswered
[Math Processing Error]
Using your answers to the previous part, determine Jeffreys prior, \pi (\theta ), in terms of \mu and \sigma. Express your answer in such a form that \pi ((1, 1)^ T) = 1.
1 answer