#7.3 Examination Marks Example
x<-c(53,15,70,73,79,48,91,20,24,91,87,15,3,78,78,62,16,15,20,32)

ADtest<-function(z) #assumes cdf has been applied
{
n<-length(z)
z<-sort(z)
-n-sum((2*(1:n)-1)*(log(z)+log(1-z[n:1])))/n
}

#Find mean, standard deviation and normal scores and apply AD test
mean(x)
sd(x)
n<-length(x)
z<-sort(pnorm((x-mean(x))/sd(x)))
ADtest(z)

#QQplot
qqnorm(x)

#Hermite-Chebyshev polynomials
g<-function(n,y)
{
if (n==0) return(1)
if (n==1) return(y)
if (n==2) return((y^2-1)/sqrt(2))
if (n==3) return((y^3-y)/sqrt(6))
if (n==4) return((y^4-6*y^2+3)/sqrt(24))
if (n==5) return((y^5-10*y^3+15*y)/sqrt(120))
if (n==6) return((y^6-15*y^4+45*y^2-15)/sqrt(720))
if (n>6) print("generate more polynomials")
}
#Vr test statistics
Vr<-function(r,x)
{
n<-length(x)
y<-(x-mean(x))/(sd(x)*sqrt((n-1)/n))
sum(g(r,y)/sqrt(n))
}
Vr(1,x)
Vr(2,x)
Vr(3,x)
Vr(4,x)
Vr(5,x)
Vr(6,x)
S4<-0
for (i in 3:6) {S4<-S4+Vr(i,x)^2}
S4
1-pchisq(S4,6-3+1)