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June 18th, 2014, 04:05 AM   #1
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Expected mean squared error

Could anyone help me to solve this exercise?
Suppose the estimation errors e_i are independent and identically distributed random variables that takes values 1 and -1 with an equal probability. X is a uniform random variable that takes on 100 possible values (P(X=x)=1/100 , x=1,2,..100).
Since x is a number that can take on different values with different probabilities, we should talk about the expected MSE, where the expectation E x is the averaging procedure over the probability distribution of x.

Calculate E_AE and E_EA with the formulas below for N=100 and tell if the relationship between E_AE and E_EA hold.
Please show me how to solve this exercise

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June 18th, 2014, 12:17 PM   #2
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Quote:
Calculate E_AE and E_EA with the formulas below for N=100 and tell if the relationship between E_AE and E_EA hold.
What relationship?

Since the e_i are independent and have mean 0, the only terms that survive in E_EA are the $\displaystyle e_i^2$ terms, so the expressions are almost the same, the difference being a factor of 1/N.
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June 18th, 2014, 01:55 PM   #3
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I try to explain better. We are talking about the problem of averaging a crowd. we say the number that a crowd of N people wants to guess is x, and each person in the crowd makes a guess yi : yi = x+ei (x) i.e. the true value plus some error ei ... then the book explains that we are interested to compare the following two quantities:
1.the average of expected mean-squared error (E_AE)
2.the expected mean-squared error of the average (E_EA)
Since x is a number that can take on different values with different probabilities, we should talk about the expected MSE, where the expectation E x is the averaging procedure over the probability distribution of x.

if the estimates yi are independent, we have E_EA=(1/N)E_AE, instead if the estimates are completely dependent then the averaged estimate is just the same as each estimate,so the error is the same and we have E_EA=E_AE
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June 19th, 2014, 01:35 PM   #4
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Mathematically you seem to be correct. I have trouble understanding what point you are trying to make.
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June 20th, 2014, 12:52 PM   #5
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I simply would understand how to solve the exercise. Calculate E_AE and E_EA with the formulas below for N=100 considering the different probabilities of X. I don't know how to apply these data in the two formulas. At the end I think E_AE and E_EA should have a value. Which is?
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June 21st, 2014, 12:13 PM   #6
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$\displaystyle E(e_i^2) = 1, E(e_ie_j) = 0$ for all i, and j ≠ i.

Net result: $\displaystyle E_{AE}=1, E_{EA} = \frac{1}{N}$
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