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Let's say I have a non-linear model model of the form

$$\mathbf{y}(\mathbf{x})=a\cdot e^{b\cdot\mathbf{x}}$$

There are 2 parameters to fit ($a$ and $b$). In a reference book (Burnham, ed. 2002, ISBN: 0-387-95364-7, p. 61 ), it is written "K = the number of estimable parameters". According to this, it should be $k$=2, no? Other references that I found seem to say otherwise: "K: The number of parameters in the model. The default K is 2, so a model with one parameter will have a K of 2 + 1 = 3" (https://www.scribbr.com/statistics/akaike-information-criterion/)

In our case, which one is correct? $k$=2, $k$=2+1, $k$=2+2? If the right answer is $k$=2+1 or $k$=2+2, what does those additional constant 1 or 2 represent?

(and additionaly $\mathbf{n}$ is the length of vector $\mathbf{x}$ (or $\mathbf{y}$) correct?).

I am aware of this question which deals with linear models

ecjb
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    The source you give (www.scribbr.com) appears to be self-contradictory. If you deploy the item "How is AIC calculated?" if will tell you that $K$ is the number of parameters. Aside from this I see no reference in support of the claim made that $K$ has a threshold value of two, to which the number of parameters is added. – F. Tusell Feb 12 '22 at 14:52
  • Thank you for your comment @F.Tusell. So for the given example, you would say that $k$ = 2. correct? – ecjb Feb 12 '22 at 22:05
  • I would say so, but see the answer by Rob Hyndman and the page quoted of Burnham & Anderson book. I fail to see the reason for considering $\sigma^2$ as an estimated parameter. I would have thought more in keeping with Akaike's original argument to take as $K$ the number of degrees of freedom used. – F. Tusell Feb 13 '22 at 15:25

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If you continue reading Burnham and Anderson, two pages later (p63) they explain that $K$ includes the variance estimate as well. So for your model, there are $K=3$ estimable parameters: $a$, $b$ and the error variance.

The quotation from the scribbr reference makes no sense at all.

Rob Hyndman
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