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How do I minimize the mean square error of a weighted sum of two estimators?

Thanks in advance!

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Since $d_1$ and $d_2$ are supposed to be unbiased, $\mathbb E[d] = \theta$ and $E[(d - \theta)^2] = \text{Var}(d)$. Since $d_1$ and $d_2$ are independent, $$\text{Var}(\alpha d_1 + (1-\alpha) d_2) = \text{Var}(\alpha d_1) + \text{Var}((1-\alpha) d_2) = \alpha^2 \sigma_1^2 + (1-\alpha)^2 \sigma_2^2$$

Now, can you minimize that?

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    Why $d_1$ and $d_2$ are independent? – 2017-02-27
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    The original version of the question said they were. – 2017-02-27
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    Oh, really! Thank you! – 2017-02-27