Question

Suppose that X and Y are random samples of observations from a population with mean μ and variance σ².

Consider the following two unbiased point estimators of μ.

A = (7/4)X - (3/4)Y    B = (1/3)X + (2/3)Y

[Give your answers as ratio (eg: as number₁ / number₂ ) and DO NOT make any cancellation]  

1.    Find variance of A. Var(A) = ? *σ²

2.    Find variance of B. Var(B) = ? *σ²

3. Efficient and unbiased point estimator for μ is ?

Answer 1

We first check the mean of the random variables. We use these properties:

The mean of a multiple of a random variable is equal to the multiple of the mean of that variable.

The mean of a sum of random variables is equal to the sum of the mean of those variables.

The mean of a difference of random variables is equal to the difference of the mean of those variables.

Thus, we have verified that all the means are the same and equal to .

For the variance, we use the following properties:

The variance of a multiple of a random variable is equal to the square of the multiple of the variance of that variable.

The variance of a sum of random variables is equal to the sum of the variance of those variables.

The variance of a difference of random variables is equal to the sum of the variance of those variables.

1. Variance of A:

2. Variance of B:

3. The efficient and unbiased point estimator for has the smallest variance. Here we have:

So, the efficient and unbiased point estimator for is B.