Question

For the probability density function f(x∣a,b) = Γ(a)b^a * x^(a−1) * e^(−x/b), the score function for b, assuming a = a₀, can be calculated as follows:

Score function for b: ∂/∂b [ln(f(x∣a₀,b))] = -a₀/b + Σ(xi/b²)

For the probability density function f(x∣a,b) = Γ(a)b^a * x^(4−1) * e^(−x/b), the asymptotic variance of the MLE b_ML, assuming a = a₀, can be found using the observed Fisher information. The observed Fisher information (I_obs) is given by:

I_obs(b) = -E[∂²/∂b² ln(f(x∣a₀,b))] = -E[-a₀/b² + Σ(xi/b³)]

Then, you can find the asymptotic variance of the MLE b_ML as Var(b_ML) = 1/I_obs(b_ML).

For the probability density function f(x∣a,b) = Γ(a)e^(−1) * x^(a−1) * e^(−x/b), the observed Fisher information of the MLE b_ML, assuming a = a₀, can be calculated similarly to the second question. It will involve taking the second derivative with respect to b and then finding the expected value based on the given distribution.

Answer 1

The score function for b and the asymptotic variance of the MLE b_ML can be calculated for different probability density functions with the given formulas. Assuming a = a₀, the score function for b can be found by taking the partial derivative of ln(f(x∣a₀,b)) with respect to b. The asymptotic variance of the MLE b_ML can be obtained using the observed Fisher information. The process involves taking the second derivative with respect to b and finding its expected value based on the given distribution.

Step-by-step explanation:

The score function for b in the probability density function f(x∣a,b) = Γ(a)b^a * x^(a−1) * e^(−x/b), assuming a = a₀, can be calculated using the formula:
Score function for b: ∂/∂b [ln(f(x∣a₀,b))] = -a₀/b + Σ(xi/b²)

The asymptotic variance of the MLE b_ML in the probability density function f(x∣a,b) = Γ(a)b^a * x^(4−1) * e^(−x/b), assuming a = a₀, can be found using the observed Fisher information which is given by:
I_obs(b) = -E[∂²/∂b² ln(f(x∣a₀,b))] = -E[-a₀/b² + Σ(xi/b³)]
The asymptotic variance of the MLE b_ML can then be calculated as: Var(b_ML) = 1/I_obs(b_ML)

The observed Fisher information of the MLE b_ML in the probability density function f(x∣a,b) = Γ(a)e^(−1) * x^(a−1) * e^(−x/b), assuming a = a₀, is calculated in a similar manner to the second question.

This involves taking the second derivative with respect to b and then finding the expected value based on the given distribution.