There is a theorem (here, Theorem 3.2.) which says: Let $x_i \sim p_i(\mu_i, \sigma_i^2)$ for $1 \leq i \leq n$ be a set of pairwise uncorrelated random variables. Consider the linear estimator $y_{n,\alpha}(x_1, ..., x_n) = \sum_{i = 1}^n \alpha_i x_i$, where $\sum_{i = 1}^n \alpha_i = 1$. Then, the variance of the estimator is minimized for (its proof can be also found in this question):
$$\alpha_i = \frac{\sigma_i^{-2}}{\sum_{j = 1}^n \sigma_j^{-2}}.$$
My question is: In another, but somehow similar context (but not minimization), I have obtained numerically, for my problem at hand, that: $$\alpha_i = \frac{\sigma_i^{-1}}{\sum_{j = 1}^n \sigma_j^{-1}}.$$ I really appreciate that people familiar with statistics could elaborate that if my obtained formula is known in other context, or what is the meaning behind that if we want to compare it with the known formula in the above theorem. In other words, if the $\alpha_i$ in the theorem minimizes the variance of the estimator, what is the interpretation of my $\alpha_i$?
EDIT
In general, where $\alpha_i = \frac{\sigma_i^{-2}}{\sum_{j = 1}^n \sigma_j^{-2}}$ minimizes the variance of the estimator, if we decrease the exponent from $-2$ to $-3$ or increase it from $-2$ to $-1$, how the variance of estimator will change? In particular, which is my main question, whether $-1$ is a special value like $-2$.
