Mathematical Foundations for Signal Processing, Communications, and Networking

Since ${\left\{w\left[n\right],z\left[n\right]\right\}}_{n=0}^{N-1}$ are i.i.d. exponential random variables, their PDF are p(w[n]) = λ exp(−λw[n]) and p(z[n]) = λ exp(−λz[n]), respectively. The likelihood function of ${\left\{x_{}{1}_{}\left[n\right],x_2\left[n\right]\right\}}_{n=0}^{N-1}$ is then given by ${\prod }_{n=0}^{N-1}p\left(w\left[n\right]\right)p\left(z\left[n\right]\right)$. Substituting (10.32) and (10.33) into the likelihood function, we obtain the expression:

$\begin{array}{l}p\left({\left\{x_1\left[n\right],x_2\left[n\right]\right\}}_{n=0}^{N-1}|\text{λ},{\theta }_1,{\theta }_0,d\right)\\ ={\text{λ}}^{2N}\exp \left\{-\text{λ}{\displaystyle \sum _{n=0}^{N-1}\left[\left(x_1\left[n\right]-t_2\left[n\right]\right){\theta }_1+\left(x_2\left[n\right]-t_1\left[n\right]\right)-2d\right]}\right\}\\ \cdot {\displaystyle \prod _{n=0}^{N-1}I\left[x_1\left[n\right]{\theta }_1-{\theta }_0-d-t_1\left[n\right]\ge 0\right]}\\ \cdot {\displaystyle \prod _{n=0}^{N-1}I\left[-t_2\left[n\right]{\theta }_1+{\theta }_0-d+x_2\left[n\right]\ge 0\right],}\end{array}$

(10.34)

where we have used the transformations θ₀ ≔ β₀/β₁ and θ₁ ≔ 1/β₁, and I[·] is the indicator function. Notice that from the invariance property, the ML estimate ...