Question

Akaike's information criterion is a measure of the goodness of fit of an estimated statistical model that accounts for both the fit quality and model complexity. One way to calculate AIC is as follows:

$\mathit{AIC}=2k + n[\ln(\mathit{RSS})]\,$

, where $k$ is the number of parameters in the model and $\mathit{RSS}$ is the residual sum of squares.

Assume two models:

$f_1(x) = ax$

and

$f_2(x) = \frac{ax}{b + cx} + d \sin(x)$

. Is the number of parameters, $k$, in both the models equal 1?

EDIT In the equations above only $x$ is a variable, while $a$, $b$, $c$ ... are constant parameters

Answer 1

The question should be: how many of the parameters need to get estimated based on the data? If three of them were somehow known independently of the data, then I say for present purposes there's only one.

The nonlinearity in $x$ in the $\sin x$ term is not a concern because $f_2(x)$ is linear in $\sin x$ and the parameter is multiplied by that. But I'm wondering whether fitting by minimizing the residual sum of square makes sense when you have nonlinearity in $x$ in the other term.

Akaike's information criterion is one of many used for this purpose that I've heard of, and I'm really not familiar with any of them except to a minor extent that of Mallows.