Standard error of estimates
- Thread starter phdmama
- Start date
- Tags estimate standard error
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
\( SD[aX] = aSD[X] \)
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