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{wiki=Power_transform\#Box–Cox_transformation}

For $y>0$, the Box–Cox transformation is
$$
T_\lambda(y)=
\begin{cases}
(y^\lambda-1)/\lambda,&\lambda\ne0,\\
\log y,&\lambda=0.
\end{cases}
$$
The value of $\lambda$ can be selected by maximizing its <profile likelihood>; $\lambda=1$ corresponds to an affine transformation of the original response, while $\lambda=0$ gives a <logarithmic transformation>.

Box–Cox transformation

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A <nuisance parameter> is a <statistical parameter> required to specify the distribution of observations but not itself the target of inference. A <profile likelihood> maximizes over it; <Bayesian model evidence> integrates over it using a proper <prior distribution>. These operations differ and need not give the same inference. A <baseline hazard> in a <Cox proportional-hazards model> is an infinite-dimensional example.

Nuisance parameter

Apply , starting from $\theta^{old}=(\mu^{old},\phi^{old},(\sigma^2)^{old})$ in the stationary parameter space. The <Gaussian AR1 bridge> gives the E-step quantities
$$
m=\mu^{old}+\frac{\phi^{old}(x_1+x_3-2\mu^{old})}{1+(\phi^{old})^2},\qquad v=\frac{(\sigma^2)^{old}}{1+(\phi^{old})^2}.
$$
Thus $\mathbb E[X_2\mid x_{obs}]=m$, $\mathbb E[X_2^2\mid x_{obs}]=m^2+v$, and $\mathbb E[X_1X_2\mid x_{obs}]=x_1m$, $\mathbb E[X_2X_3\mid x_{obs}]=mx_3$.

Let $x_2^*=m$ and $x_t^*=x_t$ otherwise. Keeping $m,v$ fixed throughout the M-step, define
$$
R(\mu,\phi)=(1-\phi^2)(x_1-\mu)^2+\sum_{t=2}^{31}\{(x_t^*-\mu)-\phi(x_{t-1}^*-\mu)\}^2+(1+\phi^2)v.
$$
The last term comes from the missing observation's conditional variance in the two adjacent innovation squares. The expected complete-data <log-likelihood> is
$$
Q(\mu,\phi,\sigma^2\mid\theta^{old})=C-\frac{31}{2}\log\sigma^2+\frac12\log(1-\phi^2)-\frac{R(\mu,\phi)}{2\sigma^2}.
$$
For fixed $\phi$, differentiating the quadratic in $\mu$ gives
$$
\mu_*(\phi)=\frac{(1-\phi^2)x_1+(1-\phi)\sum_{t=2}^{31}(x_t^*-\phi x_{t-1}^*)}{(1-\phi^2)+30(1-\phi)^2}.
$$
Also the maximizing variance is $\sigma^2=R(\mu,\phi)/31$. The M-step is therefore a one-dimensional <profile likelihood> maximization:
$$
\boxed{\phi^{new}\in\arg\max_{|\phi|<1}\left\{\frac12\log(1-\phi^2)-\frac{31}{2}\log R(\mu_*(\phi),\phi)\right\},\quad\mu^{new}=\mu_*(\phi^{new}),\quad(\sigma^2)^{new}=\frac{R(\mu^{new},\phi^{new})}{31}.}
$$
Repeat the E- and M-steps until the observed-data likelihood and parameters stabilize. Exact M-steps make that likelihood nondecreasing; different initializations help detect different local optima, and convergence alone is not a guarantee of a global maximum. At an interior local maximum, impute the missing temperature by its conditional mean $m$ evaluated at the fitted parameters, retaining $v$ as its conditional uncertainty. Replacing $X_2$ by $m$ and discarding $v$ would not implement the <expectation-maximization algorithm>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 218 / 5 / d / Solution

The graph is a <profile likelihood> for the parameter $\lambda$ of a <Box–Cox transformation>. Its maximum is close to $\lambda=0$, and the displayed likelihood interval contains zero but excludes $\lambda=1$. Since $\lambda=0$ gives a <logarithmic transformation> whereas $\lambda=1$ leaves the response unchanged up to an affine transformation, the plot supports fitting
$$
\log(\text{Paymentperpolicyyr})
$$
as the response in `model2`.

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 13J / ii / Solution

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For a parameter of interest $\psi$ and nuisance parameter $\lambda$, the profile likelihood is
$$
L_p(\psi)=\sup_\lambda L(\psi,\lambda).
$$
It compares values of $\psi$ after fitting the nuisance parameter as favourably as possible at each value.


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 Incoming links: Profile likelihood