#################################################
#MATH 5301 Nonparametric Statistics
#
#Chapter 6:  Tests of the Kolmogorov-Smirnov Type


###################################################################
#Example with the uniform distribution

#Min and Max Parameters
a=10
b=20

#Sample Size
n=1000

#Generating Data
x=runif(n,min=a,max=b)
hist(x)
sort(x)

#Empirical Cumulative Distribution Function
plot(ecdf(x),xlim=c(a-(b-a),b+(b-a)))


#Theoretical CDF Function
unif.cdf=function(x){
  return(punif(x,min=a,max=b))
}

unif.cdf(10)
unif.cdf(20)
unif.cdf(15)
unif.cdf(17)

#Plotting Theoretical Cumulative Distribution Function
x.range=seq(from=a-(b-a),
            to=b+(b-a),
            length=1000)
y.range=unif.cdf(x.range)
lines(x.range,y.range,col="blue",lwd=3)


#Kolmogorov-Smirnov Test
ks.test(x,unif.cdf)


#KS-Test Confidence Band
library(sfsmisc)
ecdf.ksCI(x)
lines(x.range,y.range,col="blue",lwd=3)


#################################################################
#Normally Distributed Data

y=rnorm(n,mean=(a+b)/2,sd=(b-a)/sqrt(12))

hist(x)
hist(y)


#Empirical Cumulative Distribution Function
plot(ecdf(y),xlim=c(a-(b-a),b+(b-a)))


#Theoretical CDF Function
norm.cdf=function(x){
  return(pnorm(x,mean=(a+b)/2,sd=(b-a)/sqrt(12)))
}

#Plotting Theoretical Cumulative Distribution Function
x.range=seq(from=a-(b-a),
            to=b+(b-a),
            length=1000)
norm.y.range=norm.cdf(x.range)
lines(x.range,norm.y.range,col="blue",lwd=3)


#Kolmogorov-Smirnov Test
ks.test(y,norm.cdf)


#KS-Test Confidence Band
library(sfsmisc)
ecdf.ksCI(y)
lines(x.range,norm.y.range,col="blue",lwd=3)



############################################################
#Comparing Uniform and Normal Data

ks.test(x,y)
wilcox.test(x,y)


#Uniform Data with NormCDF
ecdf.ksCI(x)
lines(x.range,norm.y.range,col="blue",lwd=3)

#Normal Data with UnifCDF
ecdf.ksCI(y)
lines(x.range,y.range,col="blue",lwd=3)



####################################################################
#Testing Specifically for Normality (QQ-Plots and Shapiro-Wilk Test)

x=rnorm(1000)

qqnorm(x)
shapiro.test(x)


x=runif(1000)

qqnorm(x)
shapiro.test(x)