Question

If X1,X2,…,Xn constitute a random sample from a gamma population with mean αθ and variance αθ^2, where α is a known parameter. Which of the following is a maximum likelihood estimator for parameter θ ? (a). X^2. (b). αX^2. (c). αXˉ. (d) Xˉ/α

Answer 1

The maximum likelihood estimator for parameter θ in this case is Xˉ/α.

The answer is option ⇒ (d) Xˉ/α

Step-by-step explanation:

The maximum likelihood estimator for parameter θ in this case is option (d):Xˉ/α.

To find the maximum likelihood estimator, we need to find the value of θ that maximizes the likelihood function L(θ), which is the joint probability density function of the sample.

By taking the natural logarithm of the likelihood function, we obtain the log-likelihood function:

ln L(θ) = nln(αθ) - αθ²sum(Xi) - nln(Γ(α))

To find the maximum, we take the derivative of the log-likelihood function with respect to θ and set it equal to zero:

d(ln L(θ))/dθ = n/θ - 2αθsum(Xi) = 0

Solving for θ, we get:

θ = sum(Xi)/(2nα)

Hence, the maximum likelihood estimator for θ is Xˉ/α.

The answer is option ⇒ (d) Xˉ/α