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April 10th, 2008, 08:36 PM   #1
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Unbiased estimation and moments

Hi all, hope someone could help, thks a lot

let x and y be the random variable, y = f(x) and f is a nonliner continous strictly monotonically decreasing function. Let x' be the unbiased and consistent estimation of x, generally we can't say that y'=f(x') is also an unbiased and consistent estimation of y

Now, suppose f(x) can be decomposed into nth order polynomial, f(x)= c0 + c1x + c2x^2+...+cnx^2,
taking expectation, we have E{f(x')} = c0 + c1E{x'} + c2E{x'^2} +...+ cnE{x'^n}, would this prove y' is an unbiased and consistent estimation? in other words, if x' is unbiased and consistent, will the nth order moments of x' also be unbiased and consistent?

Thank you very much for your help.
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