Uncertainty from Box–Cox estimation Consider the model
$$
Y_i^{(\lambda)} = \alpha+\beta x_i + \varepsilon_i,\qquad \varepsilon_i,\ (i=1,\ldots,n) \sim \mathrm{i.i.d.}\  N(0,\sigma^2)
$$
where
$$
\begin{align}
y_i^{(\lambda)} & =
\begin{cases}
\dfrac{y_i^\lambda-1}{\lambda(\operatorname{GM}(y))^{\lambda -1}} , &\text{if } \lambda \neq 0 \\[12pt]
\operatorname{GM}(y)\log{y_i} , &\text{if } \lambda = 0
\end{cases} \\[12pt]
\text{and } & \operatorname{GM}(y) = (y_1\cdots y_n)^{1/n}\text{ is the geometric mean.}
\end{align}
$$
As my estimate of $\lambda$, I use the value that minimizes the sum of square of residuals when $\alpha$ and $\beta$ have likewise been estimated by least squares.
In finding a confidence region for $\alpha$ and $\beta$, how should one take into account the uncertainty in the estimate of the Box–Cox parameter $\lambda$?
 A: You could bootstrap the entire process and use those results to create your confidence region for alpha and beta.
You could also use a Bayesian approach and place priors on alpha, beta, and lambda (and sigma) then work out a confidence region based on the posterior.
You could use simulations to see how often the regular confidence region (without adjusting for uncertainty in lambda) includes the true values and make adjustments based on that.
In any case, the purpose of Box-Cox transformations is not to find a "best" value of lambda and just use it blindly.  The better approach is to use the information on lambda (including the confidence interval on lambda) to suggest possible values and combine that with your (and other experts) knowledge of the science that produced the data to choose a meaningful value of lambda.  For example if the "best" value for lambda is estimated to be 0.4733268 and the 95% CI on lambda includes the value of 0.5 and there is a scientifically meaningful reason that a square root transformation would make sense then you should use the square root transformation instead of raising the data to a value like 0.4733268.  Tools like the Box-Cox transformation are tools to be used along with scientific knowledge (and common sense) not to replace it.
