Case 1: No measurement errors ( ${\sigma _b^2}= 0$)

Equations (36) can be solved in closed form if ${\sigma _b^2}= 0$. In this case,

$$\hat{\phi}_0 = \frac{c_1}{c_0} \hat{{\sigma_a^2}}_0 = c_0 \, \left[ 1 - \left(\frac{c_1}{c_0}\right)^2\right],$$ (37)

where

$$c_m = \frac{1}{N} \sum_{j=1}^{N} y_j \, y_{(j+m) \bmod N}$$ (38)

is the barrel autocorrelation of $\{y_j\}$ at lag $m$. The associated statistical uncertainties are

$$\Delta \hat{\phi}_0 = \sqrt{\frac{\hat{{\sigma_a^2}}_0}{N c_0}} \Delta \hat{{\sigma_a^2}}_0 = \hat{{\sigma_a^2}}_0 \, \sqrt{\frac{2}{N}}.$$ (39)

In the absence of measurement errors, $c_0$ and $c_1$ constitute sufficient statistics for the time series (30) and thus embody all of the information that can be extracted.