In order to estimate a population mean, , 2 surveys were conducted independently and the statistics were noted. ( are obtained). Assume that and are unbaised. For some and , the two estimates can be combined to give a better estimator:
What choice of and will minimize the variances, given that ?
Since I managed to find one equation which is: . What is another equations which allow me to minimise the variance in order to find suitable and ?
There are actually 2 questions on this (However there aren't much more clues to that)
Qn1: Find the conditions on and that make the combined estimate unbiased. I get
Qn2: What choice of and minimizes the variances, subject to the condition of unbiasedness above.
Answer I suppose: (it's true that is still 1)
I'm not sure whether is the population needed since we already know their statistics.
is the standard errors. So
With the given answer: ,
I've tried to work backward in order to attain the 2nd equation:
Subst the answer into the 1st equation:
so it's valid.
Subst the answer into the 2nd equation:
However I'm not sure why the new variance estimator becomes ?