Question

Let Y1, Y2, ..., Yn be i.i.d. random variables following a uniform distribution in the interval (0, θ), where θ is the unknown parameter. We'll consider the estimation of θ.

(a) Write down the Minimum Variance Unbiased Estimator (MVUE) for θ, and calculate the variance of this MVUE.

(b) Calculate the value of the expression: n * E[(dθ / d log fY(Y1 | θ))^2], where fY(Y1 | θ) is the probability density function of Y1.

(c) Based on the result in part (b), propose a lower bound on the variance of unbiased estimators of θ. Discuss whether this lower bound is valid for the uniform distribution, considering the limitations related to the uniform distribution.

Answer 1

The MVUE for θ is 2Yn with a variance of θ2/3n. The expression n * E[(dθ / d log fY(Y1 | θ))2] simplifies to n * θ2. The lower bound on the variance of unbiased estimators of θ is θ2, but it is not valid for the uniform distribution.

Step-by-step explanation:

(a) The minimum variance unbiased estimator (MVUE) for θ in this case is 2Yn. To calculate its variance, we can use the fact that the variance of a uniform distribution on (0, θ) is θ2/12. Therefore, the variance of the MVUE is θ2/3n.

(b) The expression n * E[(dθ / d log fY(Y1 | θ))2], where fY(Y1 | θ) is the probability density function of Y1, simplifies to n * (θ2/θ)2 = n * θ2.

(c) From part (b), we can see that the lower bound on the variance of unbiased estimators of θ is θ2. However, for the uniform distribution, this lower bound is not valid because it depends on the parameter θ, which is unknown.