Smearing retransformation

The Smearing retransformation is used in regression analysis, after estimating the logarithm of a variable. Estimating the logarithm of a variable instead of the variable itself is a common technique to more closely approximate normality. In order to retransform the variable back to level from log, the Smearing retransformation is used.

If the log-transformed variable y is normally distributed with mean

$$ f(X) $$ and variance $$ \sigma^2  $$

then, the expected value of y is given by:

$$ y = \exp(f(X))\exp(\frac{1}{2}\sigma^2). $$