Estimate Function w/ Parameter: Bias & Differentiability?

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SUMMARY

The discussion centers on the bias associated with estimating a function of a parameter using a plug-in method. It is established that bias is not always present in such estimations, particularly when dealing with linear functions like f(p) = Ap + B. The conversation also touches on the specific case of estimating e-lambda using e-x', where x' is an estimate for a Poisson random variable, indicating that this estimation is biased. The participants clarify that bias can depend on the nature of the function and the distribution involved.

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  • Understanding of plug-in estimation methods
  • Knowledge of bias in statistical estimation
  • Familiarity with Poisson random variables
  • Concept of differentiability in functions
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  • Explore the implications of differentiability on bias in estimations
  • Learn about the characteristics of Poisson distributions and their estimators
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Statisticians, data scientists, and researchers involved in statistical estimation and analysis, particularly those working with Poisson distributions and bias in estimators.

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If I am trying to estimate a function of a parameter by a plug-in method:
let's say p is a parameter then P' is an estimate of it
so then f(p) can be estimated by f(p').
So, my question is whether I will ALWAYS have a bias associated with this estimation... even if the estimate function is infinitely differentiable...
prof in class gave reasons why you have bias with Normal distribution and Bernouli but not the general case...
Thanks
 
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No, you don't ALWAYS have bias.

For example f(p) = Ap + B for any constants A and B.
 
AlephZero said:
No, you don't ALWAYS have bias.

ok, then how can I show (hint, not solution) that e-x' is not an unbiased estimate of e-lambda, where lambda is a parameter and x' it's estimation for Poisson rv
thanks
 

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