# Thread: Variance of a log function

1. ## Variance of a log function

Hello,

I would very much appreciate it if someone could explain to me the reason why:

Var[ log( 1-f(x) )] =~ [ 1/(1-f(x)) ]^2 Var(f(x))

Thank you!

2. Originally Posted by stephanosh
I would very much appreciate it if someone could explain to me the reason why:

Var[ log( 1-f(x) )] =~ [ 1/(1-f(x)) ]^2 Var(f(x))
This makes little sense. Are you sure "Var" stands for "variance" and not for some "variation"? (it looks like ${\rm Var}(f(x))=(f'(x))^2$ or ${\rm Var}(f(x))=\frac{1}{h^2}(f(x+h)-f(x))^2$ for small $h$, from your formula)

3. Hello,

I am absolutely sure it stands for variance.

4. Originally Posted by stephanosh
Hello,

I would very much appreciate it if someone could explain to me the reason why:

Var[ log( 1-f(x) )] =~ [ 1/(1-f(x)) ]^2 Var(f(x))

Thank you!
Delta method.

$Var[g(x)] \approx [ g'(x) ]^2 Var(x)$

Think of it like the Taylor expansion of the variance. What you have above is the second order expansion. Note that you get some nice cancelations in your taylor series.

See: page B-5 of

http://www.phidot.org/software/mark/.../pdf/app_2.pdf

5. Thank you Anonymous1, this is precisely what I was looking for.

6. Just got out of my Stats midterm and one of the questions was: Find the approximate variance of your MEM estimator.

I haven't used Delta in a while, and don't think I would have thought of it, if I hadn't been floating it around in my head the night before.

Thanks. You are awesome for asking this question.

7. Hah! What a coincidence.. So kind of you to be answering questions in the forum the day before your midterm!

Cheers!

8. Originally Posted by stephanosh
Hah! What a coincidence.. So kind of you to be answering questions in the forum the day before your midterm!

Cheers!
Ironic procrastination.