Can you please help me show the statement below? I am not exactly sure where to start.
Choosing constant to minimize mean square error
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statistics
1 Answers
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As a function of $c$, the quadratic $\mathrm E(S^4)c^2-2\sigma^2\mathrm E(S^2)c+\sigma^4$ is minimum at $c=\sigma^2\mathrm E(S^2)/\mathrm E(S^4)$. To compute this ratio, the simplest approach could be to compute $\mathrm E(S^2)$ and $\mathrm E(S^4)$.
For starters, would you know how to compute $\mathrm E(S^2)$? Hint: the value of $\mathrm E(S^2)$ does not depend on the sample being drawn from a normal distribution, but only on the parameters $\mu$ and $\sigma^2$.
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0I just saw this post which tremendously helped and gave me a good method for finding E(S^4): http://math.stackexchange.com/questions/72975/variance-of-sample-variance – 2011-10-16