Bias of functions defined on samples for population

[mathinator]

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!

 

[Jameson]

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.

 

[mathinator]

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!