Let's consider the following example:

1) To apply the thoerem, we first need a sufficient statistic. This is easy by the factorization theorem. Next, we need unbiased estimator for theta which is harder to find. Is there a general way to pinpoint the correct one?

In the solutions they found an unbiased estimator by first computing E(Yi^2). It looks like there is some "GUESSING" factor in here. How can we think of computing E(Yi^2) in the first place? There are many possible functions of the sufficient statistic. Why choose the particular one that they've chosen? Where to get this inspiration? Is there a systmatic way to find such unbiased esimtators?

2) What I understand about the theorem now is that it does NOT give the esimator with the smallEST variance (only smallER), but in the above example, they immediately concluded from the theorem that it is the MINIMUM variance unbiased estimator. How come?

Thanks for explaining!