Mean estimation under known variance heterogeneity

We observe $X_1,\ldots,X_N$ and consider following model: $$X_i = \theta + w_i\epsilon,\quad \epsilon \sim N(0, 1).$$ Based on above model, we want to estimate $\theta$ given $X_1,\ldots,X_N$ and $w_1, \ldots, w_N$. Let $\bar{X}$ be sample mean of $X_1,\ldots,X_N$. Of course $X_i$ and $\bar{X}$ are members of unbiased estimators of $\theta$.

I'm interested in better estimator (in some sense) than above. Are there anything?

The maximum likelihood estimator will be the weighted mean $X_w = \frac{\sum 1/w_i^2 X_i}{\sum 1/w_i^2}$.
It will be unbiased and will have the smallest variance of $1/\sum(1/w_i^2)$, i.e. it is better than a simple mean or any $X_i$
• I missed about MLE. Thanks. But How can we confirm the variance minimality of $X_w$? – Citrus Nov 25 '16 at 5:34