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

{c}

In the <expectation-maximization algorithm> for a stationary <autoregressive process of order one>, a single missing interior value has a <normal distribution> conditional on its neighbors. The E-step uses both its conditional mean and its conditional second moment, not just mean imputation. The M-step maximizes the expected stationary complete-data <log-likelihood>, including the initial-observation density and the conditional-variance contribution to the two adjacent innovation squares.

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 Incoming links: EM for a missing observation in a Gaussian AR1 process