Question

Given that random variables X1, X2, …, Xn are independent and follow a gamma distribution with a probability density function (PDF) defined as f(x∣x0, α, β), and you want to estimate β/α using the maximum likelihood method (MLE), find the MLE of β/α.

Answer 1

To estimate β/α using MLE, find MLE of β and α first, then divide MLE of β by MLE of α.

Step-by-step explanation:

To estimate the parameter β/α using the maximum likelihood method (MLE), we need to find the MLE of β and the MLE of α first.

The MLE of β is the sample mean of the random variables X1, X2, …, Xn. And the MLE of α is the solution to the equation ∑(ln(Xi))/n = ψ(α) - ln(β), where ψ(α) is the digamma function.

Once we have the MLE of β and α, we can calculate the MLE of β/α by dividing the MLE of β by the MLE of α.