Abstract: Some necessary and suffoient conditions of a non-negative quadratic form Y^\prime A Y tobe the non-negative unbiased MINQE of f^\prime \theta were proposed under the model of Y= X \beta+8 with E 8=0, E 8 \varepsilon^\prime=\theta_1 V_1+\cdots+\theta_p V_p \triangleq V_\theta \geqslant 0. Let \alpha be a prior value of \theta such that V_a>0 and M \triangleq I-X\left(\boldsymbolX^\prime X\right)-X^\prime. Also: \begingathered g:\left(M V_a M\right)^1 / 2 S Y M\left(M V_a M\right)^1 / 2 \rightarrow R^p \\ g A=\left(\operatornametr A\left(M V_a M\right)^1 / 2\left(M V_a M\right)^+ V_1\left(M V_a M\right)^+\left(M V_a M\right)^1 / 2, \cdots\right)^\prime \\ g^*: R^p \rightarrow\left(M V_a M\right)^1 / 2 S Y M\left(M V_a M\right)^1 / 2 \\ g^* a=\sum_i=1^p a_i\left(M V_a M\right)^1 / 2\left(M V_a M\right)^+ V_i\left(M V_a M\right)^+\left(M V_a M\right)^1 / 2, \text where a=\left(a_1, \cdots, a_p\right)^\prime \endgathered If M V_1 M, \cdots, M V_9 M are linearly independent and f is a relative interior point in \left\\theta: \theta_1 M V_1 M+\cdots+\theta_p M V_p M \geqslant 0\right\^\mathrmdaal, then Y^\prime\left(M V_a M\right)^1 / 2+ \hatA\left(M V_a M\right)^1 / 2^+ Y, where \hatA \in \left(M V_a M\right)^1 / 2 S Y M\left(M V_a M\right)^1 / 2, is a non-negative unbiased MINQE of f^\prime \theta \triangleq f_1 \theta_1+\cdots+ f_s \theta_0 \Leftrightarrow \hatA=\left(g^* b\right)_+, where b is a solution of the unrestricted optimization problem \min \left\\left\|g\left(g^* b\right)_+-f\right\|^2: \forall b \in \boldsymbolR^p\right\.