Standard error of estimates

#1
I have the standard error of P (which is an estimator), and W is a constant, and now have the following equation E=W*sqrt(p) to obtain E as the target estimator . How can I calculate the standard error of E?
 
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BGM

TS Contributor
#2
Can you explain the objective / variables clearer?

Do you mean you have the standard error of \( p \) (which is an estimator), and W is a constant, and now you have this equation to obtain \( E \) as the target estimator in which you would like to know the standard error?
 

BGM

TS Contributor
#4
Constant part is not difficult, as we know

\( SD[aX] = aSD[X] \)

There is no general exact result for the non-linear square root function - the result need to be calculated case by case. However you can apply the Delta method to obtain an approximate SE:

\( Var\left[w\sqrt{\hat{p}}\right] = w^2 Var\left[\sqrt{\hat{p}}\right]
\approx \left(\frac {w} {2\sqrt{E[\hat{p}]}}\right)^2 Var[\hat{p}] \)

\( \Rightarrow SD\left[w\sqrt{\hat{p}}\right] \approx
\frac {w} {2\sqrt{E[\hat{p}]}} SD[\hat{p}] \)

See, e.g.
http://en.wikipedia.org/wiki/Taylor_expansions_for_the_moments_of_functions_of_random_variables