## Appendix B Detailed Longer Proofs of Theorems, Lemmas and Corollaries
### Proofs for Section 3
\begin{split}&\underbrace{\
abla\widehat{\mathbf{J}}-\mathbf{I}_{C-1}^ {\forall}}_{\text{DEV}_{\mathbf{J}}}=\mathbf{I}_{C-1}^{\forall}+\underbrace{ \
abla\widehat{\mathbf{J}}-\mathbf{I}_{C-1}^{\forall}}_{\text{DEV}_{\mathbf{J} }}+\underbrace{\
abla\widehat{\delta\mathbf{J}}-\mathbf{I}_{C-1}^{\forall}}_{ \text{DEV}_{\mathbf{J}}}\Leftrightarrow\mathbf{I}\
abla\widehat{\mathbf{J}}= \
abla\widehat{\mathbf{J}}+\
abla\widehat{\delta\mathbf{J}}\\ &\Leftrightarrow\widehat{\mathbf{\sigma}}\
abla\widehat{\mathbf{J}}= \
abla\bigg{\{}\widehat{\mathbf{J}}+\widehat{\delta\mathbf{J}}\bigg{\}} \Leftrightarrow\widehat{\mathbf{\sigma}}\widehat{\mathbf{J}}=\widehat{\mathbf{ \langle J\rangle}+\widehat{\delta\mathbf{J}}\Leftrightarrow\widehat{\mathbf{J}} =\mathbf{\langle J\rangle}+\delta\mathbf{J}\Leftrightarrow\mathbf{J}=\mathbf{ \langle J\rangle}+\delta\mathbf{J}\end{split}  \tag{355}\]with equivalence #1 obtained by cancelling out the all-ones vectors in the two sides of the first eq., - \(\mathbf{1}_{C-1}^{\vee}=\mathbf{1}_{C-1}^{\vee}-\mathbf{1}_{C-1}^{\vee}-\mathbf{1 }_{C-1}^{\vee}\). Note that #3 is not an equivalence but it is rather a one-way implication; thus, the logical flow stems backward from the identity \(\mathbf{J}=\left\langle\mathbf{J}\right\rangle+\delta\mathbf{J}\). Derivation of the first-order statistic \(\left\langle\mathbf{IEV_{J}}\right\rangle\) in (371):  
(375)  
with #1 per first eq. in (369); #2 per linearity of the mean; #3 per \(\left\langle\delta\widetilde{\mathbf{J}}\right\rangle=0\) (perturbation RVs are ZM); #4 since \(\mathbf{IEV_{J}}\) is deterministic, given by #5 by making the substitution \(\mathbf{J}:=\left\langle\mathbf{J}\right\rangle\) in \(\mathbf{IEV_{J}}=\widetilde{\mathbf{J}}-\left\langle\widetilde{\mathbf{J}} \right\rangle=\delta_{\left\langle\mathrm{nom}\right\rangle}\widetilde{ \mathbf{J}}\) (328), yielding equality #1 in the chain  
(376)  
with #2 due to \(\widetilde{\left\langle\mathbf{J}\right\rangle}^{\mathrm{nom}}\) being deterministic, i.e., equal its own mean; #3 stems from the commutativity of the mean, \({}^{\mathrm{nom}}\), \(\widetilde{\ }\) operations; #4 stems from definition \(\delta_{\mathrm{nom}}\mathbf{J}\equiv\mathbf{J}-\mathbf{J}^{\mathrm{nom}}\) (274) by plugging in \(\mathbf{J}:=\left\langle\widetilde{\mathbf{J}}\right\rangle\) and using commutativity \(\left\langle\widetilde{\mathbf{J}}\right\rangle^{\mathrm{nom}}=\left\langle \widetilde{\mathbf{J}}\right\rangle\). Derivation of the first-order statistic \(\left\langle\mathbf{DEV_{J}}\right\rangle\) in (372):  
\[\left\langle\mathbf{DEV_{J}}\right\rangle=\left\langle\
abla\widetilde{ \mathbf{J}}-\mathbf{1}_{C-1}^{\vee}\right\rangle=\left\langle\
abla\widetilde {\mathbf{J}}\right\rangle-\mathbf{1}_{C-1}^{\vee}\right\rangle=\
abla\widetilde {\mathbf{J}}\left\langle\widetilde{\mathbf{J}}\right\rangle-\mathbf{1}_{C-1}^ {\vee}=\underset{\#4}{\mathbf{E}}\mathbf{EV_{J}} \tag{377}\]  
#1 per alt def (359) of DEV; #2 per linearity of the mean; #3 per commutation of \(\
abla\) and mean; #4 again per alt def (359) of DEV, albeit under the substitution \(\mathbf{J}:=\left\langle\mathbf{J}\right\rangle\). The Mean-Square IEV (373) and DEV (374) are derived, using the IEV and DEV variances  
\[\sigma_{\mathbf{IEV_{J}}}^{2}=\sigma_{\widetilde{\mathbf{J}}-\widetilde{ \left\langle\mathbf{J}\right\rangle}^{\mathrm{nom}}}^{2}=\sigma_{\widetilde{ \mathbf{J}}}^{2} \tag{378}\]  
\[\sigma_{\mathbf{DEV_{J}}}^{2}=\sigma_{\
abla\widetilde{\mathbf{J}}-\widetilde {\mathbf{1}}_{C-1}^{\vee}}^{2}=\sigma_{\
abla\widetilde{\mathbf{J}}}^{2}= \sigma_{\widetilde{\mathbf{J}}}^{2}\{\
abla\mathbf{C}\}^{\odot 2}\widetilde{ \mathbf{I}}^{\left\langle\cdot\right\rangle} \tag{379}\]  
with the DEV variance (expressed in units of \(\sigma_{u}^{2}\)) derived using the alternative definitions of IEV (328) and DEV (359), the invariance (A98) of the variance of any RV under a deterministic shift and the XOR expressions for evaluating \(\sigma_{\
abla\widetilde{\mathbf{J}}_{c}}^{2}-\sigma_{u}^{2}\).