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.
 

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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