# Confidence interval for the population mean using different bootstrap methods under ranked set sampling

Hello, I am trying to write code in R statistical programming language regarding the confidence interval width and coverage probability of the population mean with different bootstrap methods under ranked set sampling, but I am not sure whether the code I wrote works correctly. I share the method and the codes I wrote. It'd be great if someone could help me with this. Many thanks in advance!

Method
Step 1 : Draw the ranked set sample with set size (m) and cycle size (r) from the target population. ((sets (m) are in rows, cycles (r) are in columns))
Step 2 : To create bootstrap ranked set sample, r units from the ith row (i=1,2,…,m) are randomly selected by with replacement with probability 1/r.
Step 3 : The sample mean statistic is calculated for each bootstrap sample created.

Step 4: The calculated bootstrap sample means are sorted from smallest to largest.

Step 5 : To represent B bootstrap repetition, the value of the sorted bootstrap sample averages corresponding to (B+1)α is determined as the lower limit, and the value corresponding to (B+1)(1-α ) is determined as the upper limit of confidence interval.

Step 6 : By determining the lower and upper limits for all bootstrap repetitions, confidence interval coverage probabilities and average confidence interval width are obtained.

#R Codes
#The number of set size
m<-2

#The number of cycle size
r<-4

#The number of the simulation size
sim<-10000

#The number of times the confidence interval includes the population mean
cpn<-0
#level of significance (alpha)
alpha<-0.05
#Bootstrap Repetition size
B=2000
#Upper and Lower Limits of Confidence interval
ucl<-numeric(sim) #upper limit of confidence interval
lcl<-numeric(sim) #lower limit of confidence interval

#Target Population
x<- rnorm(1000,mean = 0,sd=1) #data from standard normal dist.
x<-as.vector(x)

for (si in 1:sim){
#Ranked Set Sampling Process
#package for Ranked Set Sampling
#Samples from a target population by using ranked set sampling method
#Ranked Set Sample (cycles (r) are in rows, sets(m) are in columns)
#Ranked Set Sample (sets (m) are in rows, cycles (r) are in columns)
matrist<-t(matris)
#The matrix for Bootstrap Ranked Set Samples
x1 <- matrix(nrow = m, ncol = r)
#Sample mean observed from Bootstrap ranked set samples.
x2 <- numeric(B)

#Method 1
for (k in 1:B) {
for (j in 1:m) {
y <- sample(matrist[j,1:r], size = r,replace = TRUE,prob=(c((1/r)*rep(1,r))))
x1[j,] <- y
}
#Bootstrap Ranked Set Sample
y2 <- as.vector(x1)
#Sample mean of Bootstrap Ranked Set Sample
meanx <- mean(y2)
#Sample means(B times) of Bootstrap Ranked Set Sample
x2[k] <- meanx
}
#The mean values of Bootstrap Ranked Set samples sorted from smallest to largest
y3 <- sort(x2)

#Lower Limit for Confidence Interval (The Value that corresponding to (B+1)α)
lcl[si] <- y3[floor((B + 1) * alpha)]
#Upper Limit for Confidence Interval (The Value that corresponding to (B+1)
(1-α))
ucl[si] <- y3[ceiling((B + 1) * (1 - alpha))]

#The coverage probability of confidence interval
if(lcl[si] <= mean(x) & mean(x) <= ucl[si])
{
cpn<- cpn+1}

print(si)
}

#The mean values of upper and lower limits of confidence interval
mean(ucl)
mean(lcl)
#Average Confidence Interval Width
ıw<-mean(ucl)-mean(lcl)
ıw

#The coverage probability of confidence interval
cp<-cpn/sim
cp

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