# davies.moment: Moments of the Davies distribution In Davies: The Davies Quantile Function

## Description

Moments of order statistics of random variables drawn from a Davies distribution

## Usage

 ```1 2 3 4 5 6 7 8``` ```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) ```

## Arguments

 `params` A three-member vector holding C, lambda1 and lambda2 `n` The number of observations `i` Return information about the i-th order statistic (ie i=1 means the smallest, i=n means the biggest) `order` The order (eg order=2 gives the square)

## Details

Function `davies.moment(n,i,order=r)` gives the r-th moment of the i-th 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 central moment of a Davies distribution

• `skewness()` gives the normalized skewness of a Davies distribution

• `kurtosis()` gives the normalized kurtosis of a Davies distribution

## Author(s)

Robin K. S. Hankin

`expected.value`, `expected.gld`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33``` ```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) ```