Sum of log normals not log normal; Monte Carlo Simulation

#1
The Problem is this:

Let X1, X2 be two normaly distributed R.V. with equal Variance sigma, expectation 0 undCorrelation Rho. I Would like to:

1) simulate X1 and X2 jointly in excel; 2)simulate exp(X1) und exp(X2)jointly in excel ; 3) simulate exp(X1) + expe(X2); 4) show, that ln(exp(X1)+exp(X2) is not normal

For that i ran 1000 simulations in excel:

1)First simulate 1000 standard normal R.V. each, Z1, Z2 with: NORM.S.INV(Rand()) | NORM.S.INV(Rand())

2) simulate X1 and X2 each 1000 time with

X1=σ* Z1 | X2= + σ* [ρ*Z1 + (1 − ρ)^2)^(1/2)*Z2]

3) simulate Y 1000 times by

Y=ln(exp(x1)+exp(X2))

Then i conducted the shapiro wilk test for y, but got a p-value of 0.15 , so can't reject H0.

Can someone tell me what i did wrong, because clearly this sum shouldnt be normal?

Thanks and greetings, götze
 

Dason

Ambassador to the humans
#2
You didn't do anything wrong. The distribution isn't normal - you're correct on that. But it isn't very different from a normal distribution. From a very rough simulation I threw together in R the power to reject the shapiro test here is approximately 0.16.