### R examples for nonparametric statistics
### Source: Hollander M, Wolfe D, Nonparametric statistical methods, 2nd edition, Wiley
### Course: APMA168-Nonparametric Statistics, Brown University
### Instructor: Konstantinos Spiliopoulos



#boots.tau is an R program that computes an asymptotically
#distribution-free confidence interval for Kendall's tau
#based on Efron's bootstrap method.  See Section 8.4.  One must specify,
#using the paired data Z_1=(X_1,Y_1), ..., (X_n,Y_n), the vector
#x=(X_1,...,X_n), the vector y=(Y_1,...,Y_n), alpha, where 1-alpha is the
#confidence coefficient, and B, the number of bootstrap
#replications.  The output contains the list of the ordered bootstrap replications as
#well as the lower and upper limits of the confidence interval.


boots.tau<-function(x,y,alpha,B)
{
  alpha<-(alpha/2)
  n<-length(x) #(x,y) is the sample values
               # 1-alpha is the confidence coefficient
   #B is the size of the bootstrap sample, e.g. B=1000
   xnew<-numeric()
   ynew<-numeric()
   taunew<-numeric()

  #Main loop that generates the bootsrtap sample
   for(i in 1:B)
   {
     a<-ceiling(10*runif(n,0,n/10))
     for(j in 1:n)
     {
      xnew[j]<-x[a[j]]
      ynew[j]<-y[a[j]]
     }
     # calculating the tau's for the samples (xnew,ynew) using the function "tau"
     taunew[i]<-tau(xnew,ynew)
   }
   taunew
   taunew<-sort(taunew)
   #Getting the quantiles
   L<-taunew[trunc((alpha)*(B+1))] 
   b<-trunc((alpha)*(B+1))     
   U<-taunew[(B+1-b)]
   #Create a list of objects containing the vector "taunew"
   # and the quantiles L and U
   #To extract them use sum.stat[[]]
   sum.stat<-list(taunew,L,U)
   sum.stat
}

tau<-function(x,y)
{
 n<-length(x)
 s<-0
 p<-0
 for(i in 1:(n-1))
 {
  for(j in (i+1):n)
  {
   a<-(x[i]-x[j])*(y[i]-y[j])
   if(a>0)
       s<-s+1
   else if(a==0)
       s<-s
   else
       s<-s-1
  }
 }
 p<-(2*s)/(n*(n-1))
 p
}

### Application to canned Tuna data of table 8.1

  Hunter <- c(44.4, 45.9, 41.9, 53.3, 44.7, 44.1, 50.7, 45.2, 60.1)
  Panel <- c( 2.6,  3.1,  2.5,  5.0,  3.6,  4.0,  5.2,  2.8,  3.8)
  example<-boots.tau(Hunter,Panel, 0.1,1000)
  hist(example[[1]])
   
###The code is from the book of Hollander-Wolfe.