Does Box-Cox parameter estimation count towards parameters for AIC? Suppose I have a regression model with e.g. 2 parameters
$y = ax + b$
But the data are non-normal so before regressing I transform both sides with Box-Cox estimation.  Thus I get two Box-Cox parameters as well, $\lambda_x$ and $\lambda_y$.
Now I want to calculate AIC for this model.  How many parameters are there?
My instinct would be that $\lambda_y$ counts as a parameter but not $\lambda_x$, because if the model is applied to forecasting $y$ from $x$, $\lambda_x$ can be estimated any time from the available $x$ but we have to remember which $\lambda_y$ to use as we don't know the $y$ we are trying to predict.
 A: Write $p_\lambda(x)$ for the Box-Cox transformation of $x$ with parameter $\lambda$, $-\infty\lt\lambda\lt\infty$.  The full model for data $(x_i,y_i)$ where the responses $(y_i)$ are viewed as a realization of a random vector $(Y_i)$ is described in the question as
$$\mathbb{E}(p_{\lambda_y}(Y_i)) = a + b\, p_{\lambda_x}(x_i).$$
That explicitly has four parameters ${a, b, p_{\lambda_y}, p_{\lambda_x}}$, all of which are identifiable provided there are at least three distinct values of $x_i$ and three distinct values of $y_i$.  According to the answers to your preceding question, you count four parameters when none of the values are established independently of the data (and therefore are estimated from the data).  If instead either (or both) $\lambda_x$ or $\lambda_y$ were established in some other way--for instance, if $\lambda_y$ were computed from a separate data set--then it would not be counted.
(Depending on distributional assumptions made about $p_{\lambda_y}(Y_i)$, there could be more parameters involved in fitting the model.  Counting them is not affected by the Box-Cox transformations.  The one-to-one property of the Box-Cox transformation indicates that any parameter that is identifiable in the absence of the transformation will remain identifiable when the transformation is applied.)
A: Why are you transforming the data?
What scale are you asking your questions on?  If it isn't the transformed scale, then there is the issue of parameterization.
An example might help.  Suppose some response $Y \sim \log N(\mu, \sigma)$, and you want to compare $Y$ across two independent populations.  Suppose further that the question of interest is, "Are the means equal in these populations?"  The obvious thing to do in this case is to analyze $\log y_{ij}$ and estimate $\mu_1 - \mu_2$.
That would not answer the question, though.  $E\{Y_i\} = \exp (\mu_i+\sigma_i)$.  What you need is a confidence interval on $\mu_1-\mu_2 + \sigma_1-\sigma_2$.
Now, if in fact you are interested in things on the transformed scale none of this applies.  It happens: chemists and environmental scientists are interested in pH (to the point that they measure pH rather than $H^+$ concentrations.)
