NuclearPhynance Archive

Is [img]/User%20Files/4008/Latex-Equation-7392.gif[/img] when x small (x stochastic) ? Do anybody has a reference on this ?

If x is constant yes.

Otherwise NO! .

Take a binomial for example.

by jensen's inequality if f(x) is convex, then E(f(x)) >= f(E(x)) (with equality iff x is knst).



 



(edit : sorry cross posted)

To pj: In the binomial case 0 or Y with Y small (I assumed x small is my post) than:

[img]/User%20Files/4008/Latex-Equation-7393.gif[/img]

so it holds in this case.

Is it always true?

Thx Jaxx. I actually know that by jensen's inequality it is not equal, but I am wondering if the aproximation is ok when x small ?

Well, when x is small, it is approximately constant.

You mean ≈ as in

e^x≈1+x?

When x constant first order aproximation on x gives [img]/User%20Files/4008/Latex-Equation-7395.gif[/img] as you said, when by Jensen inequality [img]/User%20Files/4008/Latex-Equation-7396.gif[/img]

>>>but I am wondering if the aproximation is ok when x small ?



What do you exactly by x is small? X is a random variable?



What you can do is use the Delta method which basically tells you that:



if EX converges in proba to \theta; with in your case h(x)=exp(x) and assume var(X)=\sigma^2 then:



sqrt(n) E[h(x_n)-h(\theta)] converges in distribution to N(0, \sigma^2 h'(\theta))

Or furthermore you can just taylor expand exp(x) and take the expectation of you expansion (but this assume that you know the moments of the distribution of X)