Standard Deviation: Formula (8) Approximation Explained

[daudaudaudau]

Hi all.

If I have a function V(x1,x2,x3,x4) and I want to calculate it's standard deviation when x1,x2,x3,x4 are standard normal and their variances are small, then formula (8) on this page
http://www.devicelink.com/mem/archive/99/09/003.html" is an approximation to the standard deviation. Can anyone offer me a proof or tell me where I can read more about this formula?

-Anders

 

[EnumaElish]

The "clever" part is to express the deviation in Vo, dVo, as a linear combination of the partial derivatives and deviations in the individual variables (dxi, for i=1,2,3,4). The rest follows from:

http://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables

 

[daudaudaudau]

Hi.

Hmm, I think it's the clever part I don't understand then. Must I taylor expand V and then compute the standard deviation? And also, I can't see the connection between "difference" and "variance". I'd appreciate a few formulas. Thanks.

-Anders

 

[EnumaElish]

I thought it would follow from dV = Sum[(partial V/partial xi) dxi, i=1,2,3,4]. The "d" operator is similar to "deviation" (e.g., from the mean). Let's say you have only 2 x's. Then dv = (Dv/Dx1) dx1 + (Dv/Dx2) dx2 ==> dv^2 = (Dv/Dx1)^2 dx1^2 + (Dv/Dx2)^2 dx2^2 + ignored term^* approx. equal to (Dv/Dx1)^2 dx1^2 + (Dv/Dx2)^2 dx2^2. (I've used capital D for "partial.")

In fact, you don't even need the link I posted.

^*This is the interaction term 2(Dv/Dx1)(Dv/Dx2) dx1 dx2. If you think that for a given "random draw" either of dx1 or dx2 (but not necessarily both, and you don't know which) is likely to be "very small" then you can assume 2(Dv/Dx1)(Dv/Dx2) dx1 dx2 = 0.

 

[daudaudaudau]

Okay. The only part I don't understand now is why dx is the same as a standard deviation ?

 

[EnumaElish]

Because you can choose the dx's in multiples of the standard deviation for each x. So you can choose dx = sigma(x).

See also: http://en.wikipedia.org/wiki/Variance#Approximating_the_variance_of_a_function
and http://en.wikipedia.org/wiki/Delta_method#Note (for independent x's, all Cov terms = 0).

 

[daudaudaudau]

Thank you! Wikipedia led me to http://www.stata.com/support/faqs/stat/deltam.html which really helped me.