## 4 Pairwise Clock Synchronization under Exponential Delays
### Clock Offset Estimation under Exponential Delays
#### 4.1.3 Minimum Variance Unbiased Estimator  
As discussed earlier, the MVUE is an unbiased estimator that exhibits lower variance than any other unbiased estimator. Finding the MVUE usually requires usage of the concept of sufficient statistics, the Neyman-Fisher factorization theorem and the Rao-Blackwell-Lehmann-Scheffe theorem [23] (p. 101). Let \(\mathbf{\Phi}\) denote the parameters to be estimated and \(\mathbf{z}\) represent the set of the observations; then, the Neyman-Fisher factorization theorem states that if the likelihood function \(p(\mathbf{z}|\mathbf{\Phi})\) can be factorized as:  
\[p(\mathbf{z}|\mathbf{\Phi})=g(T(\mathbf{z}),\mathbf{\Phi})h(\mathbf{z}),\]  
where \(g\) is a function depending on \(\mathbf{z}\) only through \(T(\mathbf{z})\) and \(h\) is a function depending on \(\mathbf{z}\) only, then \(T(\mathbf{z})\) is a sufficient statistic for \(\mathbf{\Phi}\). In a two-way message exchange mechanism under symmetric exponential delays, \(\mathbf{\Phi}=[d,\theta,\lambda]^{T}\) and \(\mathbf{z}=[U_{1},U_{2},\cdots,U_{N},V_{1},V_{2},\cdots,V_{N}]^{T}\). The MVUE of \(\mathbf{\Phi}\) was derived in [31] by implementing the following procedure. Express first the likelihood function in terms of the unit step function \(u[\cdot]\) as follows:  
\[p(\mathbf{z}|\mathbf{\Phi})=\lambda^{-2N}\exp\left[-\frac{1}{\lambda}\sum_{i= 1}^{N}(U_{i}+V_{i}-2d)\right]\cdot u[U_{(1)}-d-\theta]\cdot u[V_{(1)}-d+\theta] \tag{18}\]and factorize Equation (18) as:  
\[p(\mathbf{z}|\mathbf{\Phi})=g_{1}(\sum\limits_{i=1}^{N}(U_{i}+V_{i}),d,\lambda) \cdot g_{2}(U_{(1)},d,\theta)\cdot g_{3}(V_{(1)},d,\theta)\cdot h(\mathbf{z})\]  
where:  
\[g_{1}(\sum\limits_{i=1}^{N}(U_{i}+V_{i}),d,\lambda)=\lambda^{-2N}\exp\left[- \frac{1}{\lambda}\sum\limits_{i=1}^{N}(U_{i}+V_{i}-2d)\right]\]  
\[g_{2}(U_{(1)},d,\theta)=u[U_{(1)}-d-\theta],\;\;g_{3}(V_{(1)},d,\theta)=u[V_{ (1)}-d+\theta],\;\;h(\mathbf{z})=1\]  
In the above equations, \(h(\mathbf{z})\) is independent of the unknown parameters \(\mathbf{\Phi}\) and \(g_{1},g_{2},g_{3}\) are functions depending on the data only through \(\mathbf{T}=\{\sum\limits_{i=1}^{n}(U_{i}+V_{i}),U_{(1)},V_{(1)}\}\). Using the Neyman-Fisher factorization theorem, it turns out that \(\mathbf{T}\) is a sufficient statistic. Moreover, the [PERSON] theorem (p. 109 in [23]) claims that a sufficient statistic \(\mathbf{T}\) is complete if there is only one function \(c(\cdot)\) of \(\mathbf{T}\) that is unbiased, and this function leads to the MVUE, _i.e._, \(\mathbf{\Phi}_{\text{MVU}}=c(\mathbf{T})\). Therefore, the remaining task is to prove that \(\mathbf{T}\) is complete or, equivalently, that only one function of \(\mathbf{T}\) is unbiased, and to find that function. The authors in [31] employed a one-to-one transformation of \(\mathbf{T}\), denoted as \(\mathbf{T}^{\prime}=\{\sum\limits_{i=1}^{N}(U_{i}+V_{i}-U_{(1)}-V_{(1)}),U_{( 1)},V_{(1)}\}\). Then, it was shown that \(\mathbf{T}^{\prime}\) is complete by assuming that there are two functions of \(\mathbf{T}^{\prime}\) leading to unbiasedness, _i.e._, \(\mathrm{E}(c(\mathbf{T}^{\prime}))=\mathrm{E}(h(\mathbf{T}^{\prime}))= \mathbf{\Phi}\), and then, proving that, actually, \(c(\mathbf{T}^{\prime})=h(\mathbf{T}^{\prime})\). It turns out that \(\mathbf{T}\) is also complete, since the sufficient statistics are unique within one-to-one transformations [23]. To this end, what remains to prove is to find an unbiased estimator for \(\mathbf{\Phi}\) as a function of \(\mathbf{T}\), which represents the MVUE according to the [PERSON] theorem. It seems difficult to find three unbiased functions of \(\mathbf{T}\) for each of \(d,\theta\) and \(\lambda\) just by inspection. However, it can be observed that the BLUE \(\mathbf{\Phi}_{\text{BLUE-S}}\) in Equation (16) is a function of \(\mathbf{T}\) and is also unbiased. Therefore, it is concluded that the BLUE is also the MVUE:  
\[\hat{\mathbf{\Phi}}_{\text{MVUE-S}}=\left[ \begin{array}{c}\hat{d}_{\text{MVUE -S}}\\ \hat{d}_{\text{MVUE-S}}\\ \hat{\lambda}_{\text{MVUE-S}}\end{array}