## 3 Theoretical Comparisons Between Estimators  
For a novel estimator to be considered a significant contribution, it must demonstrate clear advantages over existing methodologies. A rigorous theoretical framework is essential for establishing its superiority, providing a foundation for comparative analysis. Theoretical investigations enable a precise understanding of the conditions under which the new estimator outperforms or underperforms relative to conventional approaches. This, in turn, aids practitioners in determining when to adopt the new estimator and when traditional methods remain preferable. **Lemma 1**. : _Let \(B\) be a positive definite matrix, \(\zeta\) be a vector of nonzero constants and \(\
u\) be a positive constant, then \(vB-\zeta\zeta^{\prime}>0\) if and only if \(\zeta^{\prime}B^{-1}\zeta<\
u\)[33]._  
**Definition 1**. : _Let \(\hat{\theta}_{1}\), \(\hat{\theta}_{2}\) be two estimators of \(\theta\), the estimator \(\hat{\theta}_{1}\) is superior to the estimator \(\hat{\theta}_{2}\) in the sense of MMSE criterion, if and only if_  
\[\Delta\big{(}\hat{\theta}_{1},\hat{\theta}_{2}\big{)}=MMSE\big{(}\hat{\theta }_{1}\big{)}-MMSE\big{(}\hat{\theta}_{2}\big{)}>0\]  
**Lemma 2**. : _Let \(\hat{\theta}_{1}\), \(\hat{\theta}_{2}\) be two estimators of \(\theta\), \(\hat{\theta}_{1}=A_{1}y\), and \(\hat{\theta}_{2}=A_{2}y\). Assume that \(\psi=cov\big{(}\hat{\theta}_{1}\big{)}-cov\big{(}\hat{\theta}_{2}\big{)}>0\), where \(cov\big{(}\hat{\theta}_{1}\big{)}\), \(cov\big{(}\hat{\theta}_{2}\big{)}\) represent the covariance matrices of the estimators \(\hat{\theta}_{1}\), \(\theta_{2}\), respectively. Then,_  
\[ \begin{array}{c}\Delta\big{(}\hat{\theta}_{1},\hat{\theta}_{2}\big{)}=MMSE \big{(}\hat{\theta}_{1}\big{)}-MMSE\big{(}\hat{\theta}_{2}\big{)}=cov\big{(} \hat{\theta}_{1}\big{)}+a_{1}a_{1}^{{}^{\prime}}-cov\big{(}\hat{\theta}_{2} \big{)}-a_{2}a_{2}^{{}^{\prime}}\\ \Delta\big{(}\hat{\theta}_{1},\hat{\theta}_{2}\big{)}=\psi+a_{1}a_{1}^{{}^{ \prime}}-a_{2}a_{2}^{{}^{\prime}}>0\end{array} \]  
_if and only if_  
\[a_{2}^{\prime}\big{(}\psi+a_{1}a_{1}^{\prime}\big{)}a_{2}<1\]  
_where \(a_{1},a_{2}\) denote the bias vectors of the estimators \(\hat{\theta}_{1}\), \(\hat{\theta}_{2}\), respectively [33; 34]._  
**Theorem 1**. : _The estimator \(\hat{\alpha}^{PIG-MkLE}\) is superior to the estimator \(\hat{\alpha}^{PIG-MkLE}\) in the sense of MMSE criterion, i.e., \(MMSE\big{(}\hat{\alpha}^{PIG-MkLE}\big{)}-MMSE\big{(}\hat{\alpha}^{PIG-MkLE} \big{)}>0\) if and only if_  
\[ \begin{array}{c}\mathrm{bias}\big{(}\hat{\alpha}^{PIG-MkLE}\big{)}^{\prime }\Big{[}\hat{\omega}\Lambda^{-1}-\hat{\omega}\Lambda^{-1}_{k^{+}}\Lambda_{k^{ -}}\Lambda^{-1}_{k^{+}}\Lambda\Lambda^{-1}_{k^{+}}\Lambda_{k^{-}}\Lambda^{-1}_ {k^{+}}\Big{]}bias\big{(}\hat{\alpha}^{PIG-MkLE}\big{)}<1\end{array} \]  
Proof. : The difference between \(\mathrm{cov}\big{(}\hat{\alpha}^{PIG-MkLE}\big{)}\) and \(\mathrm{cov}\big{(}\hat{\alpha}^{PIG-MkLE}\big{)}\) is as follows:  
\[ \begin{array}{c}\mathrm{cov}\big{(}\hat{\alpha}^{PIG-MLE}\big{)}-\mathrm{ cov}\big{(}\hat{\alpha}^{PIG-MkLE}\big{)}=\hat{\omega}\Lambda^{-1}-\hat{ \omega}\Lambda^{-1}_{k^{+}}\Lambda_{k^{-}}\Lambda_{k^{+}}^{-1}\Lambda\Lambda^{- 1}_{k^{+}}\Lambda_{k^{-}}\Lambda^{-1}_{k^{+}}\\ \mathrm{cov}\big{(}\hat{\alpha}^{PIG-MkLE}\big{)}-\mathrm{cov}\big{(}\hat{ \alpha}^{PIG-MkLE}\big{)}=\hat{\omega}\ \mathrm{diag}\bigg{\{}\frac{1}{\lambda_{j}}-\frac{\lambda_{j} \big{(}\lambda_{j}-k\big{)}^{2}}{\big{(}\lambda_{j}+k\big{)}^{2}}\bigg{\}}_{j= 1}^{q}\end{array} \]  
Since \(\mathrm{bias}\big{(}\hat{\alpha}^{PIG-MkLE}\big{)}\mathrm{ Bias}\big{(}\hat{\alpha}^{PIG-MkLE}\big{)}^{{}^{\prime}}\) is non-negative definite, then the difference matrix \(\mathrm{cov}\big{(}\hat{\alpha}^{PIG-MkLE}\big{)}-\mathrm{cov}\big{(}\hat{ \alpha}^{PIG-MkLE}\big{)}\) is positive definite if and only if  
\[ \begin{array}{c}\big{(}\lambda_{j}+k\big{)}^{4}>\lambda_{j}^{2}\big{(} \lambda_{j}-k\big{)}^{2},\ \mathrm{for}\ k>0\ \mathrm{.}\end{array}