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) |
itemdavies.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)
itemM
gives the r-th moment for n=i=1
itemmu
gives the first moment of a Davies distribution (ie the mean)
itemvariance
gives the second {em central} moment of a Davies
distribution
itemskewness
gives the normalized skewness of a Davies distribution
itemkurtosis
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)