Estimate Function w/ Parameter: Bias & Differentiability?

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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
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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