#(1)
x=c(5.7,5.5,4.3,5.9,5.2,5.6,5.8,5.1)
y=c(5.0,4.5,4.2,5.4,4.4)
m=length(x)
n=length(y)
N=m+n

#(a) Wilcoxonuv test - presny
wilcox.test(x,y,exact=TRUE,cor=FALSE)
R=rank(c(x,y)) 
t=sum(R[(m+1):N]) #testova statistika

X=combn(N,n)
K=length(X[1,])

W=apply(X,2,sum)
a=table(W)/K        #pravdepodobnostni funkce

2*sum(a[1:7]) #presna p-hodnota


#(b) Wilcoxonuv test - asymptoticky
wilcox.test(x,y,exact=FALSE,cor=FALSE)
W=sum(R[(m+1):N])
EW=n*(N+1)/2
DW=m*n*(N+1)/12
t=abs((W-EW)/sqrt(DW))  #testova statistika
2*(1-pnorm(t))          #asymptoticka p-hodnota


#(c) van der Waerdenuv test - presny
t=sum(qnorm(R[(m+1):N]/(N+1))) #testova statistika


W=apply(qnorm(X/(N+1)),2,sum)
a=table(W)/K        #pravdepodobnostni funkce

2*sum(W<=t)/K  #presna p-hodnota


#(d) van der Waerdenuv test - asymptoticky
library(snpar)

data=c(x,y)
group=c(rep(0,m),rep(1,n))
ns.test(data,group)

S=sum(qnorm(R[(m+1):N]/(N+1)))
DS=m*n/N/(N-1)*sum(qnorm((1:N)/(N+1))^2)
t=abs(S/sqrt(DS))      #testova statistika
2*(1-pnorm(t))         #asymptoticka p-hodnota


#(e) medianovy test - presny
t=sum(sign(R[(m+1):N]/(N+1)-1/2)) #testova statistika

W=apply(sign(X/(N+1)-1/2),2,sum)
a=table(W)/K   #pravdepodobnostni funkce

2*sum(a[1:3])  #presna p-hodnota


#(f) medianovy test - asymptoticky
R=rank(c(x,y))
S=sum(sign(R[(m+1):N]/(N+1)-1/2))
DS=m*n/N
t=abs(S/sqrt(DS))   #testova statistika
2*(1-pnorm(t))      #asymptoticka p-hodnota


#(g+h) Kolmogorovuv-Smirnovuv test
z <- cumsum(ifelse(order(c(x,y)) <= m, 1/m, -1/n))
z      #hodnoty rozdilu emp. distribucnich funkci v bodech Z_(1),...,Z_(N)
D=max(abs(z))
D

ks.test(x,y,exact=F)
t=sqrt(m*n/(m+n))*D  #testova statistika
2*exp(-2*t^2)        #priblizna p-hodnota

#(2)
m=10
n=15
N=n+m
opak=10000
zam1=0
zam2=0
zam3=0
zam4=0
zam5=0


set.seed(1234)
for (i in 1:opak){
x=rnorm(m)
y=rnorm(n)+0.5
data=c(x,y)
group=c(rep(0,m),rep(1,n))
if (t.test(x,y)$p.val<0.05) zam1=zam1+1
if (wilcox.test(x,y)$p.val<0.05) zam2=zam2+1
if (ns.test(data,group)$p<0.05) zam3=zam3+1

R=rank(data)
S=sum(sign(R[(m+1):N]/(N+1)-1/2))
T=abs(S/sqrt(m*n/(N-1)))
if (abs(T)>qnorm(0.975)) zam4=zam4+1

if (ks.test(x,y)$p.val<0.05) zam5=zam5+1

}

zam1/opak*100  #t-test
zam2/opak*100  #Wilcoxonuv test
zam3/opak*100  #van der Waerdenuv test
zam4/opak*100  #medianovy test
zam5/opak*100  #Kolmogorovuv-Smirnovuv test