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For $X\sim N(\mu 1,C)$ with a known positive-definite matrix $C$, the maximum-likelihood estimator of $\mu$ is $\hat{\mu }=(1^T{C}^{-1}X)/(1^T{C}^{-1}1)$. It is unbiased with variance $(1^T{C}^{-1}1{)}^{-1}$, attaining the Cramer-Rao bound. It is also the best linear unbiased estimator. Pairwise admissible correlation coefficients alone do not ensure that $C$ is a valid covariance matrix; the entire matrix must be symmetric and a positive semidefinite matrix. Inverse-based formulas require positive definiteness.

For independent unbiased estimators $X_i$ of a common mean with known positive variances $v_i$, the linear unbiased estimator with smallest variance has weights $w_i=v_i^{-1}/{\sum }_j{v}_j^{-1}$ and variance $({\sum }_j{v}_j^{-1}{)}^{-1}$. Unbiasedness requires ${\sum }_i{w}_i=1$. Minimizing ${\sum }_i{v}_i{w}_i^2$ by a Lagrange multiplier gives the weights, and the strictly positive diagonal Hessian matrix proves a unique global minimum. This conclusion requires no normality assumption.

Linearity of expectation gives ${\mathbb{E}}_{\mu }[\hat{\mu }]=({\alpha }_1+{\alpha }_2)\mu$. Thus a linear unbiased estimator for every value of the distance modulus satisfies $\boxed{{\alpha }_1+{\alpha }_2=1}$. Requiring unbiasedness at the single value $\mu =0$ would not impose this constraint; an unbiased estimator must work throughout the parameter space.

For this model, the Gauss-Markov theorem states that $\hat{\beta }$ is the unique minimum-variance linear unbiased estimator of $\beta$: ifis unbiased for every $\beta$, then $\mathrm{var}(\tilde{\beta })\ge \mathrm{var}(\hat{\beta })$.

Indeed,so unbiasedness is equivalent to ${\sum }_i{a}_i{x}_i=1$. WriteThe constraint becomes ${\sum }_i{c}_i{x}_i=0$. Since the errors are independent with common variance ${\sigma }^2$,Equality holds exactly when every $c_i=0$, which gives $a_i=x_i/S_{xx}$ and $\tilde{\beta }=\hat{\beta }$.