## 4 Pairwise Clock Synchronization under Exponential Delays
### Clock Offset Estimation under Exponential Delays
#### 4.1.2 Best Linear Unbiased Estimator  
In the statistical signal processing field, the minimum mean square error (MSE) is commonly selected as the criterion to measure the performance of estimators. The MSE of an estimator \(\hat{\theta}\) is defined as:  
\[\text{MSE}(\hat{\theta})=\text{var}(\hat{\theta})+[\text{bias}(\hat{\theta})] ^{2} \tag{13}\]  
where \(\text{var}(\hat{\theta})\) stands for the variance of \(\hat{\theta}\) and \(\text{bias}(\hat{\theta})\stackrel{{\triangle}}{{=}}\text{E}( \hat{\theta})-\theta\).  
In general, the MVUE is referred to as the \"optimal\" estimator, since it minimizes the variance in the class of unbiased estimators. However, it frequently occurs that the MVUE is difficult or impossible to be derived or might not even exist [23]. For such scenarios, a suboptimal estimator is required. A common approach is to constrain the estimator to be linear in terms of the observations and to find the linear estimator that is unbiased and achieves the minimum variance. Such an estimator is referred to as the BLUE. The BLUE of the clock offset under asymmetric exponential delays was first derived by [PERSON] and [PERSON] in [30] using the order statistics of the observations \([U_{(1)},U_{(2)},\cdots,U_{(N)},V_{(1)},V_{(2)},\cdots,V_{(N)}]^{T}\) where \(U_{(1)}\leq U_{(2)}\leq\cdots\leq U_{(N)}\) and \(V_{(1)}\leq V_{(2)}\leq\cdots\leq V_{(N)}\). Specifically, the BLUE is formulated as a linear combination of the order statistics:  
\[T=\sum_{i=1}^{N}a_{i}U_{(i)}+\sum_{i=1}^{N}b_{i}V_{(i)} \tag{14}\]  
The variance of estimator \(T\) and the unbiasedness condition are further expressed in terms of \(a_{i}\) and \(b_{i}\). Using Lagrange multipliers, it can be shown that the minimum of the variance subject to the unbiasedness condition is achieved when \(b_{i}=-a_{i}\) with \(a_{1}=1/2+1/(2N)\) and \(a_{i}=-1/[2N(N-1)]\) for \(2\leq i\leq N\). Thus, the BLUE of the clock offset admits the form:  
\[\hat{\theta}_{\text{BLUE-A}}=\frac{N(U_{(1)}-V_{(1)})-(\overline{U}-\overline{ V})}{2(N-1)} \tag{15}\]  
On the other hand, the BLUE of the clock offset under symmetric exponential delays can also be derived following the aforementioned steps, and it is expressed as:  
\[\hat{\theta}_{\text{BLUE-S}}=\frac{U_{(1)}-V_{(1)}}{2}\]  
Using statistical signal processing techniques, [31] later re-derived the joint BLUEs for the fixed delay, the clock offset and the mean of exponential delays under symmetric and asymmetric exponential delays, as shown in Equations (16) and (17), respectively.  
\[\left[\begin{array}{c}\hat{d}_{\text{BLUE-S}}\\ \hat{\theta}_{\text{BLUE-S}}\\ \hat{\lambda}_{\text{BLUE-S}}\end{array}\right]=\frac{1}{2(N-1)}\left[\begin{array} []{c}N(U_{(1)}+V_{(1)})-(\overline{U}+\overline{V})\\ (N-1)(U_{(1)}-V_{(1)})\\ N(\overline{U}+\overline{V})-N(U_{(1)}+V_{(1)})\end{array}\right] \tag{16}\]  
\[\left[\begin{array}{c}\hat{d}_{\text{BLUE-A}}\\ \hat{\theta}_{\text{BLUE-A}}\\ \hat{\lambda}_{\text{1 BLUE-A}}\\ \hat{\lambda}_{\text{2 BLUE-A}}\end{array}\right]=\frac{1}{2(N-1)}\left[\begin{array} []{c}N(U_{(1)}+V_{(1)})-(\overline{U}+\overline{V})\\ N(U_{(1)}-V_{(1)})-(\overline{U}-\overline{V})\\ 2N(\overline{U}-U_{(1)})\\ 2N(\overline{V}-V_{(1)})\end{array}\right] \tag{17}\]  
These results were further used as a prerequisite to find the MVUE of the clock offset, a result that will be presented in the next sub-section.