MHB Bias of functions defined on samples for population

AI Thread Summary
The discussion centers on whether the estimate $$\frac{1}{n} \sum_{i}^{n} f(Xi)$$ is always an unbiased estimator for $$\frac{1}{N} \sum_{i}^{N} f(xi)$$ for any function f. Participants agree that not all functions f yield unbiased estimates, citing the sample variance correction as a key example. It is noted that certain sample parameters may require adjustments to ensure unbiasedness for population parameters. The conversation emphasizes the need for clarity regarding the function f and its implications on bias. Overall, the consensus is that counter-examples exist, indicating that the unbiasedness of the estimate is not guaranteed for arbitrary functions.
mathinator
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Let X1, · · · , Xn be a simple random sample from some finite population of values {x1, · · · xN }.
Is the estimate $$\frac{1}{n} \sum_{i}^{n} f(Xi)$$ always unbiased for $$\frac{1}{N} \sum_{i}^{N} f(xi)$$ no matter what f is?My thinking: I don't think all f's are unbiased, because not all sample parameters (ex: variance, or s^2) are unbiased for the population parameter (unless they are corrected for finite population sampling). I am confused if I am interpreting the question correctly, i.e f refers to parameters we can kind about the population :(

Thank you for your help in advance!
 
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Hi mathinator,

Welcome to MHB! :)

Yep I fully agree with your thought process. The sample variance correction is a great example of how this won't work for any arbitrary $f$. I think one counter-example is sufficient to wrap this problem, unless more detail is explicitly specified.
 
Jameson said:
Hi mathinator,

Welcome to MHB! :)

Yep I fully agree with your thought process. The sample variance correction is a great example of how this won't work for any arbitrary $f$. I think one counter-example is sufficient to wrap this problem, unless more detail is explicitly specified.

Thank you for your response!
 
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