Its kind of super ridiculous in NONMEM

I encountered the fact that NONMEM sometime deals with mathematically different error distributions as if they are the same. Don't worry, no one would notice it!

It happened when I trie to compare the models with different error distributions as follows: \begin{align} Y &= \mbox{IPRED} \times (1 + \mbox{EPS}(1)) \nonumber \\ Y &= \mbox{IPRED} \times \mbox{EXP}(\mbox{EPS}(1)), \nonumber \end{align} and they gave me the same objective function value(OFV). Why is that? Because it is obvious that
\begin{align} 1 + \mbox{EPS}(1) \ne \mbox{EXP}(\mbox{EPS}(1)). \nonumber \end{align}

The answer might be that NONMEM uses first order approximation. Especially for this case, the first order Taylor expansion seems to be used. The Taylor series of a function \(f(x)\) around a value \(a\) is expressed as: \begin{align} f(x) \approx f(a)+ \frac{f'(a)}{1\!}(x-a)+ \frac{f''(a)}{2\!}(x-a)^2+ \frac{f'''(a)}{3\!}(x-a)^3+\dots, \nonumber \end{align} and it is referred to as Maclaurin expansion if \(a=0\).

With the case above, the Maclaurin expansion of the function \(\mbox{EXP}(\mbox{EPS}(1))=e^{\epsilon_1}:=e^x\) around \(0\) can be expressed as: \begin{align} e^{x} &= e^{0} + e^{0}(x-0)+e^{0}(x-0)^2+e^{0}(x-0)^3+\dots \nonumber \\ &= 1 + x + x^2 + x^3 + \dots. \label{eq:taylor} \end{align} Assume \(x \approx 0\) so that \(e^{x}=1 + x\), where the first order Taylor approximation holds. The only way to excuse the fact that NONMEM deals with mathematically different error distributions as mentioned before is to use Eq.(\ref{eq:taylor}). It would be true as long as the first-order conditional estimation method (FOCE) is used as optimization method. In other word, NONMEM might not take the case where the value of the error is big into consideration.