| davies.moment {Davies} | R Documentation |
Moments of order statistics of RVs drawn from a Davies distribution
davies.moment(n=1 , i=1 , order=1 , params) M(order,params) mu(params) expected.value(n,i,params) expected.value.approx(n,i,params) variance(params) skewness(params) kurtosis(params)
params |
A three-member vector holding~C , lambda1 and~lambda2 |
n |
The number of observations |
i |
Return information about the ith order statistic (ie i=1 means the smallest, i=n means the biggest) |
order |
The order (eg order=2 gives the square) |
Function davies.moment() gives the rth moment of the ith order statistic of
n observations. The following aliases are just newbie wrappers with
n=i=1 (ie moments of one observation from a Davies
distribution):
M() gives the r-th moment for n=i=1
mu() gives the first moment of a Davies distribution
(ie the mean)
variance() gives the second {em central} moment of a Davies
distribution
skewness() gives the normalized skewness of a Davies
distribution
kurtosis() gives the normalized kurtosis of a Davies
distribution
Robin K. S. Hankin
params <- c(10,0.1,0.1)
davies.moment(n=100,i=99,2,params) # ie the second moment of the 99th smallest
# observation of 100 drawn from a Davies
# distribution with parameters p
mean(rdavies(1e6,params))-mu(params)
#now reproduce the S-K graph:
f <- function(x,y){c(skewness(c(1,x,y)),kurtosis(c(1,x,y)))}
g <- function(j,vector,pp,qq=1){points(t(sapply(vector,f,y=j)),type="l",col="black",lty=qq)}
vector <- c((0:300)/100 , (0:300)/10000 , seq(from=3,to=10,len=100))
vector <- sort(unique(vector))
plot(t(sapply((0:10)/10,f,y=0)),xlim=c(-3,3),ylim=c(0,10),type="n",xlab="skewness",ylab="kurtosis")
g(0.001,vector,"red",qq=1)
g(0.01,vector,"yellow",qq=2)
g(0.02,vector,"green",qq=3)
g(0.05,vector,"blue",qq=4)
g(0.1 ,vector,"purple",qq=5)
g(0.14,vector,"black",qq=6)
x <- seq(from=-3,to=3,len=30)
points(x,x^2+1,type="l",lwd=2)
leg.txt <- expression(lambda[2]==0.001,lambda[2]==0.01,lambda[2]==0.02,lambda[2]==0.05,lambda[2]==0.1,lambda[2]==0.14)
legend(-1.1,10,leg.txt,col="black",lty=1:6)