Random variable $X$ follows a symmetric and unkown distribution.
$\lbrace x_n \rbrace$ are a large (~$10^6$) sample drawn from $X$
Expectation $a = E[X]$ is known.
Consider the taylor expansion of $f(y) = \frac{1}{y}$ around $a$:
$g(y) = a^{-1} - a^{-2}(y - a) + a^{-3}(y - a)^{2}$
I guess $\sum_{n}g(x_{n})$ is not an unbiased approximation of $\sum_{n}f(x_{n})$ because the error of the Taylor expansion is not antisymmetric around $a$ for some distribution. By around I mean ±0.5 around the expectation.
Is it possible to correct the bias?
What if I am trying to fix a multi-variable Taylor expansion?
Error of the Taylor expansion when $X$ follows a uniform distribution over interval [0.5, 1.5]:
Minimize:
$\int_{0.5}^{1.5} g(b, c, d, y)^{2} dy + \int_{0}^{0.5} (g(b, c, d, 1 - y) - g(a, b, c, 1 + y))^{2} dy $
where $g(b, c, d, y) = 1/y - [b + c(y - 1) + d(y - 1)^{2}] $
Result:
b, c, d = 0.99265897, -1.18379802, 1.271714