Statistic that is unbiased for Sigma^2 + ^2

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In summary: Expert summarizerIn summary, the poster is seeking a statistic that is unbiased for δ^2 + μ^2 in terms of the sample sum U and sample sum of squares W. The correct statistic can be found by substituting the estimators for the sample mean and sample variance into the expression for W and manipulating it to get it in terms of U and W. The expected value of the statistic should equal δ^2 + μ^2.
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cimmerian
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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?
 
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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!
 

FAQ: Statistic that is unbiased for Sigma^2 + ^2

1. What is an unbiased statistic for Sigma^2 + ^2?

An unbiased statistic for Sigma^2 + ^2 is a measure of variability that does not consistently overestimate or underestimate the true value of the population variance.

2. How is an unbiased statistic for Sigma^2 + ^2 calculated?

An unbiased statistic for Sigma^2 + ^2 is typically calculated using the sample variance formula, which involves taking the sum of squared deviations from the mean divided by the sample size minus 1.

3. Why is it important to have an unbiased statistic for Sigma^2 + ^2?

Having an unbiased statistic for Sigma^2 + ^2 is important because it allows for accurate estimation of the population variance, which is a crucial measure of variability in a dataset. Biased statistics can lead to incorrect conclusions and inaccurate predictions.

4. Can any statistic be considered unbiased for Sigma^2 + ^2?

No, not all statistics are unbiased for Sigma^2 + ^2. It is important to carefully select and calculate a statistic that meets the criteria for unbiasedness in order to accurately estimate the population variance.

5. What are some common examples of unbiased statistics for Sigma^2 + ^2?

Some common examples of unbiased statistics for Sigma^2 + ^2 include the sample variance, the sample standard deviation, and the median absolute deviation from the mean.

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