I fitted a gaussian mixture to my financial data. The data can be found
here: http://uploadeasy.net/upload/32xzq.rar
I look at the density with
plot(density(dat),col="red",lwd=2)
this has a skew of
library(e1071)
skewness(dat)
-0.1284311
Now, I fit a gaussian mixture according to:
f(l)=πϕ(l;μ1,σ21)+(1−π)ϕ(l;μ2,σ22)
with:
datnormalmixturegaussian<-normalmixEM(dat,lambda=c(0.3828,(1-0.3828)),k=2,fast=TRUE)
save the values with:
pi<-datnormalmixturegaussian$lambda[1]
mu1<-datnormalmixturegaussian$mu[1]
mu2<-datnormalmixturegaussian$mu[2]
sigma1<-datnormalmixturegaussian$sigma[1]
sigma2<-datnormalmixturegaussian$sigma[2]
the values are:
pi = 0.383
mu1= -0.00089
mu2= 0.00038
sigma1= 0.0123
sigma2= 0.02815
Plot the single densities and the mixture:
plot.new()
xval<-seq(-0.06,0.06,length=1000)
mixturedensity<-pi*dnorm(xval,mu1,sigma1)+(1-pi)*dnorm(xval,mu2,sigma2)
plot(xval,mixturedensity,type="l",lwd=2,col="black",cex.axis=1.2,cex.lab=1.2,main="Single
univariate normal densities and mixture
density",xlab="Loss",ylab="Density",ylim=c(0,36))
curve(dnorm(x,mu1,sigma1),add=TRUE,lty=2,col="darkgreen")
curve(dnorm(x,mu2,sigma2),add=TRUE,lty=2,col="blue")
legend("topright",
legend=c("Mixture density\n","normal distribution\n of stable
market
regime\n","normal distribution\n of crash market regime\n"),
bty = "n",lwd=2, cex=1,
col=c("black","blue","darkgreen"), lty=c(1,2,2))
One can see, that both single distributions have a mean of almost zero,
wherease one has a high volatility
and the other a low volatility. The normal distribution 1, the green one
with the high peak has the parameters
mu1= -0.00089 and sigma1=0.0123 and occurs (this is pi from output of
normalmixEM) with a probability of 0.383.
The normal distribution 2 with the smaller peak and the higher volatility
has the parameters
mu2=0.00038 and sigma2=0.02815 and a probability of 1-0.383.
I imagine the generating of the mixture density as follows:
We have a distribution which is quite probable (1-0.383) and has
mu2=0.00038. If the mixture density
is done, we "add" a second distribution which is a bit shifted to the
left
(this one occurs with a probability of
0.383 and has a negative mean). Since the distribution we add lies a bit
more to the left I would expect, that the
mixture density has a negative skew, since the left tail of the resulting
mixture will be heavier?
I control this with:
skewness(mixturedensity)
which gives a positive skew of 0.7065.
Now my question is: Why? I would expect a negative skew, since I thought
the mixture density will have a a fatter left tail, since we add to the
probable distribution with positive mean a second distribution which is a
bit shifted to the left?
[[alternative HTML version deleted]]
