\longrightarrow 0} g(y(\bm{a}_i)) = \ln(y(\bm{a}_i) + \lambda_2)$.

Box and Cox give the following theorem concerning the choice of the parameters $\lambda_1$ and $\lambda_2$. Assume that each observation $y(\bm{a}_{i})$ is a realization of the random variable $G_i$, and that each transformed observation, $g(y(\bm{a}_{i}))$ is a realization of the random variable $G'_{i}$. Furthermore assume that $G'_{1}, \ldots, G'_{N}$ are independent and identically normal, i.e. assume that for all $n$, $$\begin{aligned}
  G'_{i} \sim N(\mu_\mathrm{t}^{\phantom{2}}, \sigma_\mathrm{t}^2).
  \label{Box_Cox_assumption}\end{aligned}$$ for some mean, $\mu_\mathrm{t}^{\phantom{2}}$, and variance $\sigma_\mathrm{t}^2$. Then the joint distribution of the (untransformed) observations has PDF $$\begin{aligned}
    \begin{split}
    &f_{(G_{1}, \ldots, G_{N})}(y(\bm{a}_{1}), \ldots, y(\bm{a}_{N}); \lambda_1, \lambda_2) \\
    &= \prod_{i = 1}^{N} \dfrac{1}{\sqrt{2 \pi \sigma_\mathrm{t}^2}} \exp \left( \dfrac{\sum_{i = 1}^{N} (g(y(\bm{a}_{i})) - \mu_\mathrm{t})^2}{2 \sigma_\mathrm{t}^2} \right) J(y(\bm{a}_{i}); \lambda_1, \lambda_2),
    \end{split}\end{aligned}$$ where the Jacobian $$\begin{aligned}
    J(y(\bm{a}_{i}); \lambda_1, \lambda_2)
    &= \left| \dfrac{{\operatorname{d}}g(y(\bm{a}_i))}{{\operatorname{d}}y(\bm{a}_i)} \right|\\
    &= (y(\bm{a}_i) + \lambda_2)^{(\lambda_1 - 1)}.\end{aligned}$$ Thus the log-likelihood of the parameters $\lambda_1$ and $\lambda_2$ is given by $$\begin{aligned}
    \begin{split}
        &\ln L_{(\lambda_1, \lambda_2)}(\lambda_1, \lambda_2; y(\bm{a}_{1}), \ldots, y(\bm{a}_i)) = - \frac{N}{2} \ln \sigma_\mathrm{t}^2 - \frac{N}{2}\log 2 \pi \\
        & \quad - \frac{1}{2\sigma_\mathrm{t}^2}\sum_{i = 1}^{N} (g(y(\bm{a}_i)) - \mu_\mathrm{t})^2 + (\lambda_1 - 1) \sum_{i = 1}^{N} \ln(y(\bm{a}_i) + \lambda_2)
    \end{split}
    \label{eq:Box_Cox_likelihood}\end{aligned}$$ where we may substitute for $\mu_\mathrm{t}^{\phantom{2}}$ and $\sigma_\mathrm{t}^2$ their maximum-likelihood 