Expectation of Covariance Estimate

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SUMMARY

The discussion focuses on calculating the expectation of the covariance estimate, specifically addressing the need to separate instances where i equals j for the terms E[XiYj]. The participants emphasize the importance of incorporating mean terms, suggesting the addition of +X_bar - X_bar to the expression. Furthermore, they recommend completing the square to derive E[(X-X_bar)(Y-Y_bar)], which will help in identifying the bias in the covariance estimate. The goal is to transform the biased estimate into an unbiased one through these mathematical manipulations.

PREREQUISITES
  • Understanding of covariance and its mathematical properties
  • Familiarity with expectation notation in statistics
  • Knowledge of mean terms and their role in statistical calculations
  • Ability to manipulate algebraic expressions, including completing the square
NEXT STEPS
  • Research the properties of covariance and its expectation in statistical theory
  • Learn about bias in statistical estimates and methods to correct it
  • Study the concept of completing the square in algebraic expressions
  • Explore advanced statistical techniques for estimating covariance in multivariate distributions
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Statisticians, data analysts, and anyone involved in statistical modeling or estimation who seeks to understand and correct bias in covariance estimates.

brojesus111
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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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Is this the next step? What's after that if so?

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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.
 

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