Question

f(x;θ)=θx​e−x2/(2θ) (a) It can be shown that E(X2)=2θ. Use this fact to construct an unbiased estimator of u based on ∑Xi2​ (and use rules of expected value to show that it is unbiased). (b) Find the maximum likelihood estimator (θ^MLE​) of θ ?

Answer 1

To construct an unbiased estimator of u based on ∑Xi^2, use T = ∑Xi^2/2n. To find the maximum likelihood estimator (MLE) of θ, solve for θ in the likelihood function f(x; θ) = θx&e^(-x^2/2θ) by taking the derivative of the log-likelihood function and setting it equal to zero. The MLE is θ^MLE = ∑Xi^2/2n.

Step-by-step explanation:

(a) Constructing an unbiased estimator of u based on ∑Xi^2:

Given that E(X^2) = 2θ, we want to find an estimator of θ based on ∑Xi^2. Let's define the estimator as T = ∑Xi^2/2n.

To show that T is unbiased, we need to find E(T) and show that it is equal to θ.

E(T) = E(∑Xi^2/2n)

= ∑E(Xi^2/2n) (using linearity of expectation)

= ∑(E(Xi^2)/2n) (since Xi's are independent)

= ∑(2θ/2n)

= (2θ/2n)*n

= θ

Therefore, T = ∑Xi^2/2n is an unbiased estimator of θ.

(b) Finding the maximum likelihood estimator (MLE) of θ:

To find the MLE of θ, we need to find the value of θ that maximizes the likelihood function f(x; θ) = θx&e^(-x^2/2θ).

By taking the derivative of the log-likelihood function and setting it equal to zero, we can find the value of θ that maximizes the likelihood.

Solving for θ, we find θ^MLE = ∑Xi^2/2n.