Expectation of Covariance Estimate

In summary, the conversation is about trying to take the expectation of the covariance estimate and finding a way to make it unbiased. The next step is to incorporate mean terms and complete the square in order to find the bias.
  • #1
brojesus111
39
0
So I'm trying to take the expectation of the covariance estimate.

I'm stuck at this point. I know I have to separate the instances where i=j for the terms of the form E[XiYj], but I'm not quite sure how to in this instance.

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The answer at the end should be biased, and I'm trying to find a way to make it unbiased. But first tings first, I have to simplify the above.
 
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  • #2
Is this the next step? What's after that if so?

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  • #3
Hey brojesus111 and welcome to the forums.

I think you will have to incorporate the mean terms by putting something like + X_bar - X_bar.

Also given your expression, another that comes to find is try and complete the square in the way of getting E[(X-X_bar)(Y-Y_bar)] by matching this expression with the one you have been given.

The difference between the two will give the bias.
 

Related to Expectation of Covariance Estimate

1. What is the expectation of a covariance estimate?

The expectation of a covariance estimate is the average value that we would expect to obtain if we were to repeatedly estimate the covariance between two variables using a given dataset. It is also known as the mean of the covariance estimate.

2. How is the expectation of a covariance estimate calculated?

The expectation of a covariance estimate can be calculated by taking the average of all possible covariance estimates that can be obtained from a given dataset. This is done by summing up all the covariance estimates and dividing by the total number of estimates.

3. Why is the expectation of a covariance estimate important?

The expectation of a covariance estimate is important because it provides a measure of the accuracy of the covariance estimate. A lower expectation indicates a more accurate estimate, while a higher expectation indicates a less accurate estimate.

4. How does sample size affect the expectation of a covariance estimate?

The expectation of a covariance estimate is affected by sample size. As the sample size increases, the expectation of the covariance estimate decreases, indicating a more accurate estimate. This is because a larger sample size provides more information and reduces the variability of the estimate.

5. Can the expectation of a covariance estimate be negative?

Yes, the expectation of a covariance estimate can be negative. This can occur if the relationship between the two variables is weak or negative, and the random sampling process results in a negative covariance estimate. However, in most cases, the expectation of a covariance estimate is expected to be positive since most variables have a positive correlation.

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