Monday, July 2, 2012
* Jensen's inequality
* If f is convex then the Expected value of f(x) is greater that f of the expected value of x.
* If f'(x) >= 0 E(f(x)) >= f(E(x))
* The reverse with concave.
* Likewise if f'(x) <= 0 E(f(x)) <= f(E(x))
* This is easy to show:
* Let's start with 1000 observations
set obs 100
* This will make it so that x is always positive making derivatives of functions easy to interpret.
* f(x)=x^2 -> f''(x)=2 > 0 thus convex
gen fx = x^2
* The expected value is approximated by the mean of x
local fEx = (r(mean))^2
di "So: f(E(x)) ~ " string(`fEx',"%9.2f")
di "E(f(x)) ~ " string(r(mean),"%9.2f") " is greater than f(E(x)) ~ " string(`fEx',"%9.2f")
* Thus Jensen's inequality works!
* It is a very helpful property in probability theory.
* However, the medians function passes through monotonic functions (completely unrelated to Jensen's inequality)
qui sum fx, detail
local medfx = r(p50)
qui sum x, detail
local medx = r(p50)
di "med(f(x)) ~ " string(`medfx',"%9.2f") " is approximately the same as f(med(x)) ~ " string(`medx'^2,"%9.2f")