Statistic that is unbiased for Sigma^2 + ^2

[cimmerian]

μ

Homework Statement

Let X1...Xn be a random sample of size n from a normal distribution, Xi~N(μ, sigma^2), and define U = ƩXi and W = ƩXi^2.

Find a statistic that is unbiased for δ^2 + μ^2 in terms of U and W.

Homework Equations

xbar (sample mean) = Ʃxi/n
S^2 (sample variance)(Ʃxi^2 + n*xbar^2)/(n-1)

The Attempt at a Solution

The real answer is W/n. However I am getting (Wn^2 - U^2)/n^2(n-1) from plugging in the estimators. This worked for the parameter 2μ - 5δ^2. What do I need to do?

 

[mighty2000]

Thank you for your question. It seems like you are on the right track with your attempt at a solution. However, there are a few things that need to be corrected in your approach.

Firstly, the statistic you are looking for should be in terms of both U and W. In your attempt, you only have W in the numerator and U in the denominator. Additionally, the statistic should be unbiased for δ^2 + μ^2, not 2μ - 5δ^2. This means that the expected value of the statistic should equal δ^2 + μ^2.

To find the correct statistic, you can start by substituting the estimators for the sample mean and sample variance into the expression for W. This will give you an expression in terms of U and n. Then, using the fact that E(Xi) = μ and E(Xi^2) = δ^2 + μ^2, you can manipulate the expression to get it in terms of U and W.

I hope this helps. If you need further assistance, please let me know. Good luck with your homework!