= Bias correction after an inverse transformation
{title2=$\mathbb E[g(Z)]\simeq g(m)+\tfrac12 g''(m)\sigma^2$}
A transformed-response regression estimates the conditional mean $m$ on the transformed scale. Inverting that mean does not generally estimate the original-scale mean, because nonlinear transformations do not commute with <expected values>. For a smooth inverse $g$ and centered transformed error of variance $\sigma^2$, a second-order <Taylor expansion> gives the displayed local correction. For $g(z)=z^q$ at positive $m$, it is $m^q+\tfrac12q(q-1)m^{q-2}\sigma^2$. The variance is the response's residual variance, not the sampling variance of the estimated mean. This approximation requires errors small enough that the inverse is meaningful over the relevant range; it is not an exact moment formula for arbitrary transformed noise.
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