estimates, $$\begin{aligned}
    \hat{\mu}_{\mathrm{t}} &= \frac{1}{N} \sum_{i = 1}^N g(y(\bm{a}_i)) \text{ and}\\
    \hat{\sigma}_\mathrm{t}^2 &= \dfrac{1}{N} \sum_{i = 1}^N (g(y(\bm{a}_i)) -
    \hat{\mu}_{\mathrm{t}})^2.\end{aligned}$$ The MLE of the transformation parameters has an associ ated distribution, but it is common to consider it known, and not propagate this uncertainty through the subsequent analysis. It is also common to round off the value of $\lambda_1$ to the nearest half-integer, e.g. to use one of the values $2, 1, 1 / 2, 0, -1 / 2, -1,$ or $-2$ (the square, identity, square root, logarithm, reciprocal square root, reciprocal, or reciprocal square) which give the transformation a ready interpretation. The natural transformation of the likelihood (\[eq:likelihood\]) is to its logarithm together with an offset that ensures this logarithm is defined when the likelihood is zero. In this case the likelihood of the offset, $\lambda_2$, is strictly decreasing, and its optimization will fail. In such a case we may use some arbitrary small value for $\lambda_2$, say the smallest non-zero member of the sample.

We must be careful about the direction of the implication in this theorem. It is not the case that the maximum-likelihood estimates of $\lambda_1$ and $\lambda_2$ ensure that the transformed variables are normal. We 