> #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)
[1] 48.5
> sd(x)
[1] 30.7973
> n<-length(x)
> z<-sort(pnorm((x-mean(x))/sd(x)))
> ADtest(z)
[1] 0.9312634
> 
> #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)
[1] 9.107298e-18
> Vr(2,x)
[1] 5.516421e-16
> Vr(3,x)
[1] 0.03612702
> Vr(4,x)
[1] -1.465615
> Vr(5,x)
[1] -0.0461477
> Vr(6,x)
[1] 1.89065
> S4<-0
> for (i in 3:6) {S4<-S4+Vr(i,x)^2}
> S4
[1] 5.726018
> 1-pchisq(S4,6-3+1)
[1] 0.2205650
>